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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4190 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,26,30)(20,27,28,23)(25,31,32,29)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,23,24,27)(19,25,26,32)(22,29,30,31)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(2,7)(6,9)(8,14)(13,15)(19,26)(22,30)(25,32)(29,31)(36,44)(39,48)(41,43)(45,47);(3,10)(6,9)(11,16)(13,15)(18,24)(22,30)(23,27)(29,31)(35,42)(38,46)(41,43)(45,47);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=hi,c^2=h,f^2=g,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, 8006734);