Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,26)(19,28,29,23)(20,25,33,42)(22,24,35,27)(30,40,32,37)(31,36,43,38)(34,39,46,41)(44,47,45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,26,28)(20,32,33,30)(22,31,35,43)(24,38,27,36)(25,40,42,37)(34,45,46,44)(39,47,41,48);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,33)(22,35)(23,36)(25,41)(26,27)(28,38)(37,48)(39,42)(40,47)(44,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,21,33)(18,25,26,42)(19,30,29,32)(22,34,35,46)(23,37,28,40)(24,39,27,41)(31,44,43,45)(36,47,38,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,29)(20,33)(24,27)(31,43)(34,46)(37,40)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6967768);