Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | |
1681 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)(3,6)(4,5)(7,8)(17,18,23,36,22,27,19,24)(20,30,28,25,33,38,32,41)(21,34,37,39,35,26,29,43)(31,45,42,48,46,44,47,40);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,36)(20,28,33,32)(21,29,35,37)(25,38,41,30)(26,39,34,43)(31,42,46,47)(40,45,48,44);(9,11)(10,13)(12,16)(14,15)(17,20,22,33)(18,25,27,41)(19,28,23,32)(21,31,35,46)(24,38,36,30)(26,40,34,48)(29,42,37,47)(39,45,43,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21)(18,26)(19,29)(20,31)(22,35)(23,37)(24,39)(25,40)(27,34)(28,42)(30,44)(32,47)(33,46)(36,43)(38,45)(41,48);(2,5)(6,8)(10,14)(13,15)(21,35)(26,34)(29,37)(31,46)(39,43)(40,48)(42,47)(44,45)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7219628);