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)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(1,2)(3,10)(4,12)(5,6)(7,9)(8,14)(11,15)(13,16)(18,21)(24,26)(25,29)(31,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(2,6)(7,10)(8,12)(15,16)(17,19,23,20)(18,24,21,26)(22,28,30,27)(25,32,29,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,3,5,9)(2,7,6,10)(4,13,14,11)(8,16,12,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,4,5,14)(2,8,6,12)(3,11,9,13)(7,15,10,16)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,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, 8323965);