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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,23,24,20)(21,25,30,29)(27,31,32,28)(34,40)(35,43)(36,45)(42,47);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,34,39,40)(35,38,43,48)(36,46,45,41)(37,42,44,47);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,35)(34,38)(36,44)(37,45)(39,43)(40,48)(41,42)(46,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32)(33,36,39,45)(34,41,40,46)(35,44,43,37)(38,42,48,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,37,39,44)(34,42,40,47)(35,45,43,36)(38,41,48,46);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,38,39,48)(34,43,40,35)(36,47,45,42)(37,41,44,46)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343842);