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 | |
4224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,24)(20,28)(21,25)(22,26)(23,29)(27,31)(30,32)(33,34)(35,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(2,3)(7,9)(8,10)(15,16)(17,19)(18,21)(20,27)(22,25)(23,31)(24,26)(28,30)(29,32)(35,42)(38,39)(43,48)(46,47);(2,3)(7,9)(8,10)(15,16)(20,29)(23,28)(27,32)(30,31)(33,35,37,42)(34,38,41,39)(36,43,45,48)(40,46,44,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,23)(19,27)(21,31)(22,29)(24,30)(25,32)(26,28)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,46)(45,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19)(18,24)(20,27)(21,26)(22,25)(23,30)(28,31)(29,32)(33,36,37,45)(34,40,41,44)(35,43,42,48)(38,46,39,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,24)(18,25,26,19)(20,30,29,31)(23,27,28,32)(33,34)(35,38)(36,44)(37,41)(39,42)(40,45)(43,47)(46,48)Polycyclic presentation:
2^9|c^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625795);