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 | |
3997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,13)(15,16)(17,18,24,23)(19,27,26,22)(20,30,21,25)(28,31,29,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,14)(11,15)(13,16)(17,19,24,26)(18,22,23,27)(20,28,21,29)(25,31,30,32)(34,38)(39,41)(40,45)(46,47);(2,6)(7,10)(8,12)(15,16)(17,20,24,21)(18,25,23,30)(19,28,26,29)(22,31,27,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,21,24,20)(18,25,23,30)(19,29,26,28)(22,31,27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,5,14)(2,8,6,12)(3,11,9,13)(7,15,10,16)(17,22,24,27)(18,26,23,19)(20,31,21,32)(25,29,30,28);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,23,24,18)(19,27,26,22)(20,25,21,30)(28,31,29,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8323962);