Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491117 |
| 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 | |
381 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 381 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)(3,19,26,66)(4,27,31,83)(5,32,37,56)(6,15,39,58)(7,14,38,59)(8,17,9,16)(11,43,49,22)(12,50,53,29)(13,35,57,36)(20,67,71,101)(21,72,77,106)(23,48,78,47)(24,65,25,46)(28,84,88,110)(30,89,42,60)(33,54,93,92)(34,61,41,55)(40,62,96,52)(44,70,102,81)(45,80,104,74)(51,94,109,97)(63,98,112,69)(64,75,113,76)(68,100,119,128)(73,90,99,103)(79,114,123,105)(82,95,107,87)(85,108,86,91)(111,121,126,124)(115,122,127,118)(116,125,117,120);(1,3,7,23,8,24,39,43,10,26,38,78,9,25,6,22)(2,11,15,46,16,47,59,66,18,49,58,65,17,48,14,19)(4,20,40,79,30,70,50,98,31,71,96,123,42,81,29,69)(5,21,41,80,35,75,93,99,37,77,34,74,36,76,33,73)(12,44,60,105,52,101,83,112,53,102,89,114,62,67,27,63)(13,45,61,106,56,103,92,113,57,104,55,72,32,90,54,64)(28,68,97,124,87,118,86,117,88,119,94,121,95,122,85,116)(51,100,84,125,108,127,107,126,109,128,110,120,91,115,82,111);(1,4)(2,12)(3,20)(5,28)(6,29)(7,40)(8,30)(9,42)(10,31)(11,44)(13,51)(14,27)(15,60)(16,52)(17,62)(18,53)(19,63)(21,68)(22,69)(23,79)(24,70)(25,81)(26,71)(32,91)(33,85)(34,94)(35,87)(36,95)(37,88)(38,96)(39,50)(41,97)(43,98)(45,100)(46,105)(47,101)(48,67)(49,102)(54,82)(55,110)(56,108)(57,109)(58,89)(59,83)(61,84)(64,111)(65,114)(66,112)(72,120)(73,116)(74,121)(75,118)(76,122)(77,119)(78,123)(80,124)(86,93)(90,115)(92,107)(99,117)(103,127)(104,128)(106,125)(113,126);(1,5,8,35,10,37,9,36)(2,13,16,56,18,57,17,32)(3,21,24,75,26,77,25,76)(4,28,30,87,31,88,42,95)(6,33,7,41,39,93,38,34)(11,45,47,103,49,104,48,90)(12,51,52,108,53,109,62,91)(14,54,15,61,59,92,58,55)(19,64,46,106,66,113,65,72)(20,68,70,118,71,119,81,122)(22,73,23,80,43,99,78,74)(27,82,60,84,83,107,89,110)(29,85,40,97,50,86,96,94)(44,100,101,127,102,128,67,115)(63,111,105,125,112,126,114,120)(69,116,79,124,98,117,123,121)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,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, 398314);