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 | |
3955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,5,7)(4,12,14,8)(11,15,13,16)(20,28)(24,29)(27,32)(30,31)(33,34)(35,38)(36,41)(37,42)(39,43)(40,45)(44,46)(47,48);(1,3,6,5)(2,7,9,10)(4,13,14,11)(8,16,12,15)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(34,35)(36,43)(40,42)(44,48);(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,4,6,14)(2,8,9,12)(3,11,5,13)(7,15,10,16)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8403782);