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 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(17,18,22,27)(19,26,30,23)(20,29,31,24)(21,25,32,28)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(1,2,6,3)(4,9,11,7)(5,10,13,8)(12,14,16,15)(17,19,22,30)(18,23,27,26)(20,28,31,25)(21,29,32,24)(34,39)(36,41)(37,46)(42,48);(2,3)(4,11)(8,10)(12,16)(20,31)(21,32)(24,29)(25,28)(33,35,38,43)(34,36,39,41)(37,42,46,48)(40,44,45,47);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,20,22,31)(18,24,27,29)(19,28,30,25)(21,26,32,23)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,22,32)(18,25,27,28)(19,29,30,24)(20,26,31,23)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8680152);