Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #34597 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4061 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)(3,10,13,7)(4,8,14,5)(11,15,16,12)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,6,13)(2,7,9,10)(4,11,14,16)(5,15,8,12)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,4)(2,8)(3,11)(5,9)(6,14)(7,12)(10,15)(13,16)(18,23)(22,28)(24,30)(29,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(2,9)(4,14)(7,10)(11,16)(19,25)(22,28)(26,31)(29,32)(36,44)(39,48)(41,43)(45,47);(1,5,6,8)(2,4,9,14)(3,12,13,15)(7,16,10,11)(18,23)(22,28)(24,30)(29,32)(35,42)(38,46)(41,43)(45,47);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)Polycyclic presentation:
2^9|a^2=gi,b^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8007501);