Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C2 | C4×D32 |
| prime radical | #13741 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,14,10,17,9,44)(3,18,21,67,24,70,45,69)(4,12,26,16,8,42,28,65)(5,29,32,86,36,89,35,99)(7,38,37,101,41,104,40,54)(11,46,48,74,52,108,51,20)(13,55,57,96,59,95,107,33)(15,61,60,27,63,85,103,31)(19,66,23,78,43,68,79,125)(22,76,75,126,77,116,105,121)(25,80,82,88,34,97,84,127)(30,62,90,123,93,109,100,124)(39,56,102,118,106,120,72,119)(47,73,64,91,50,114,110,71)(49,98,111,128,113,81,112,92)(53,115,58,94,87,117,122,83);(1,3)(2,11)(4,19)(5,30)(6,21)(7,39)(8,43)(9,45)(10,24)(12,47)(13,56)(14,48)(15,62)(16,64)(17,52)(18,29)(20,33)(22,31)(23,26)(25,81)(27,77)(28,79)(32,90)(34,98)(35,100)(36,93)(37,102)(38,49)(40,72)(41,106)(42,50)(44,51)(46,55)(53,116)(54,112)(57,118)(58,121)(59,120)(60,123)(61,75)(63,109)(65,110)(66,80)(67,86)(68,97)(69,99)(70,89)(71,83)(73,115)(74,96)(76,87)(78,88)(82,92)(84,128)(85,105)(91,94)(95,108)(101,111)(103,124)(104,113)(107,119)(114,117)(122,126)(125,127);(1,4,6,26,10,8,9,28)(2,12,14,16,17,42,44,65)(3,19,21,23,24,43,45,79)(5,25,32,82,36,34,35,84)(7,27,37,85,41,31,40,61)(11,47,48,64,52,50,51,110)(13,53,57,58,59,87,107,122)(15,54,60,38,63,101,103,104)(18,66,67,78,70,68,69,125)(20,71,46,73,74,91,108,114)(22,72,75,39,77,102,105,106)(29,80,86,88,89,97,99,127)(30,81,90,92,93,98,100,128)(33,83,55,115,96,94,95,117)(49,109,111,124,113,62,112,123)(56,116,118,121,120,76,119,126)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=fg,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60791);