Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3751 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32)(34,39)(36,47)(37,49)(40,44)(45,55)(46,56)(54,61)(62,64);(2,3)(4,11)(8,10)(12,16)(18,23)(19,25)(24,30)(26,31)(34,39)(35,43)(36,47)(42,51)(46,56)(48,60)(52,57)(62,64);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32)(33,34,38,39)(35,44,43,40)(36,41,47,50)(37,42,49,51)(45,57,55,52)(46,58,56,53)(48,54,60,61)(59,64,63,62);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,35)(34,40)(36,45)(37,46)(38,43)(39,44)(41,52)(42,53)(47,55)(48,59)(49,56)(50,57)(51,58)(54,62)(60,63)(61,64);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)(33,36)(34,41)(35,45)(37,48)(38,47)(39,50)(40,52)(42,54)(43,55)(44,57)(46,59)(49,60)(51,61)(53,62)(56,63)(58,64);(33,37,38,49)(34,42,39,51)(35,46,43,56)(36,48,47,60)(40,53,44,58)(41,54,50,61)(45,59,55,63)(52,62,57,64)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022560);