Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153261 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13381 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 3 generators on 128 points:
(1,2,8,14)(3,15,21,44)(4,22,26,80)(5,28,32,11)(6,33,37,78)(7,38,40,89)(9,20,45,17)(10,46,49,25)(12,53,56,36)(13,57,59,31)(16,63,23,81)(18,70,34,98)(19,43,76,113)(24,84,29,79)(27,87,86,119)(30,92,95,97)(35,58,102,118)(39,55,91,123)(41,105,47,114)(42,108,54,121)(48,85,51,90)(50,93,117,101)(52,94,88,120)(60,82,106,65)(61,100,111,73)(62,69,103,72)(64,115,67,77)(66,122,83,109)(68,110,74,96)(71,116,99,107)(75,125,104,128)(112,127,124,126);(1,3)(2,9)(4,23)(5,20)(6,34)(7,19)(8,21)(10,47)(11,44)(12,54)(13,43)(14,45)(15,28)(16,26)(17,32)(18,37)(22,77)(24,64)(25,65)(27,83)(29,67)(30,68)(31,72)(33,96)(35,71)(36,100)(38,103)(39,104)(40,76)(41,49)(42,56)(46,82)(48,106)(50,116)(51,60)(52,61)(53,73)(55,109)(57,69)(58,124)(59,113)(62,89)(63,79)(66,86)(70,97)(74,95)(75,91)(78,110)(80,115)(81,84)(85,114)(87,125)(88,111)(90,105)(92,98)(93,127)(94,121)(99,102)(101,126)(107,117)(108,120)(112,118)(119,128)(122,123);(1,4,8,26)(2,10,14,49)(3,16)(5,29,32,24)(6,35,37,102)(7,39,40,91)(9,41)(11,51,28,48)(12,55,56,123)(13,58,59,118)(15,60)(17,67)(18,71)(19,75)(20,64)(21,23)(22,78,80,33)(25,53,46,36)(27,57,86,31)(30,93,95,101)(34,99)(38,50,89,117)(42,109)(43,112)(44,106)(45,47)(52,119,88,87)(54,122)(61,125)(62,116)(63,70)(65,73)(66,72)(68,126)(69,83)(74,127)(76,104)(77,110)(79,97,84,92)(81,98)(82,100)(85,94,90,120)(96,115)(103,107)(105,108)(111,128)(113,124)(114,121)Polycyclic presentation:
2^9|a^2=i,c^2=dh,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=eg,f^c=fi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60458);