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 | |
4138 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)(3,9)(4,12)(5,6)(7,11)(8,14)(10,15)(13,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(2,6)(7,9)(8,12)(15,16)(17,19)(18,20)(21,27)(22,23)(24,26)(25,28)(29,32)(30,31)(35,44)(39,41)(45,47)(46,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,20,22,24)(18,23,26,19)(21,29,30,28)(25,27,32,31)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,30)(31,32)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19)(18,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,37,38,40)(34,42,43,36)(35,46,44,48)(39,47,41,45)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8900196);