Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #19258 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
380 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,10,18)(3,19,26,68)(4,27,31,84)(5,32,38,58)(6,15,40,60)(7,14,39,61)(8,17,9,16)(11,44,50,22)(12,51,54,29)(13,36,59,37)(20,69,73,104)(21,74,33,95)(23,49,79,48)(24,67,25,47)(28,85,90,113)(30,91,43,62)(34,56,97,94)(35,63,42,57)(41,64,100,53)(45,72,105,82)(46,81,55,76)(52,98,111,101)(65,102,115,71)(66,77,112,78)(70,103,86,108)(75,106,96,92)(80,116,124,107)(83,99,109,89)(87,110,88,93)(114,122,126,125)(117,123,127,120)(118,128,119,121);(1,3,7,23,8,24,40,44,10,26,39,79,9,25,6,22)(2,11,15,47,16,48,61,68,18,50,60,67,17,49,14,19)(4,20,41,80,30,72,51,102,31,73,100,124,43,82,29,71)(5,33,42,76,36,78,97,75,38,21,35,81,37,77,34,96)(12,45,62,107,53,104,84,115,54,105,91,116,64,69,27,65)(13,55,63,74,58,106,94,66,59,46,57,95,32,92,56,112)(28,86,101,122,89,123,88,118,90,70,98,125,99,120,87,119)(52,108,85,121,110,117,109,114,111,103,113,128,93,127,83,126);(1,4)(2,12)(3,20)(5,28)(6,29)(7,41)(8,30)(9,43)(10,31)(11,45)(13,52)(14,27)(15,62)(16,53)(17,64)(18,54)(19,65)(21,70)(22,71)(23,80)(24,72)(25,82)(26,73)(32,93)(33,86)(34,87)(35,98)(36,89)(37,99)(38,90)(39,100)(40,51)(42,101)(44,102)(46,103)(47,107)(48,104)(49,69)(50,105)(55,108)(56,83)(57,113)(58,110)(59,111)(60,91)(61,84)(63,85)(66,114)(67,116)(68,115)(74,121)(75,118)(76,122)(77,120)(78,123)(79,124)(81,125)(88,97)(92,127)(94,109)(95,128)(96,119)(106,117)(112,126);(1,5,8,36,10,38,9,37)(2,13,16,58,18,59,17,32)(3,21,24,77,26,33,25,78)(4,28,30,89,31,90,43,99)(6,34,7,42,40,97,39,35)(11,46,48,92,50,55,49,106)(12,52,53,110,54,111,64,93)(14,56,15,63,61,94,60,57)(19,66,47,95,68,112,67,74)(20,70,72,120,73,86,82,123)(22,75,23,81,44,96,79,76)(27,83,62,85,84,109,91,113)(29,87,41,101,51,88,100,98)(45,103,104,127,105,108,69,117)(65,114,107,128,115,126,116,121)(71,118,80,125,102,119,124,122)Polycyclic presentation:
2^9|a^2=i,b^2=eg,d^2=g,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,h^a=hi,d^b=di,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398317);