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 | |
4227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,23,28)(19,24,30,31)(20,25,21,26)(22,27,32,29)(33,34,39,44)(35,42,37,40)(36,41,45,47)(38,43,48,46);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,23,30)(18,24,28,31)(20,27,21,29)(22,25,32,26)(35,37)(36,45)(40,42)(41,47);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,20,23,21)(18,25,28,26)(19,29,30,27)(22,31,32,24)(33,35,39,37)(34,40,44,42)(36,43,45,46)(38,47,48,41);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,21,23,20)(18,26,28,25)(19,29,30,27)(22,31,32,24)(33,36)(34,41)(35,43)(37,46)(38,40)(39,45)(42,48)(44,47);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,30)(26,32)(28,31)(33,37,39,35)(34,42,44,40)(36,43,45,46)(38,47,48,41);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,22,23,32)(18,27,28,29)(19,25,30,26)(20,24,21,31)(33,38)(34,43)(35,41)(36,40)(37,47)(39,48)(42,45)(44,46)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8290992);