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 | |
4116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,10,15)(4,13,9,5)(11,16,14,12)(18,23)(19,25)(21,29)(28,32)(33,34,39,43)(35,44,46,40)(36,37,47,41)(38,48,45,42);(1,3,6,10)(2,7,8,15)(4,11,9,14)(5,12,13,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,35)(34,40)(36,38)(37,42)(39,46)(41,48)(43,44)(45,47);(2,8)(3,10)(4,9)(12,16)(19,25)(20,27)(26,31)(28,32)(33,36,39,47)(34,41,43,37)(35,38,46,45)(40,48,44,42);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,19)(18,20)(21,26)(22,25)(23,27)(24,28)(29,31)(30,32)(33,36)(34,41)(35,45)(37,43)(38,46)(39,47)(40,42)(44,48);(1,4,6,9)(2,5,8,13)(3,11,10,14)(7,12,15,16)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,37,39,41)(34,36,43,47)(35,42,46,48)(38,44,45,40);(1,5)(2,9)(3,12)(4,8)(6,13)(7,14)(10,16)(11,15)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,38,39,45)(34,42,43,48)(35,36,46,47)(37,44,41,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034634);