Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,28)(29,41)(32,39)(33,44)(34,37)(35,40)(36,43)(38,42)(45,48)(46,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,35,43)(24,37,39,31)(25,36,40,41)(33,45,46,42)(38,47,48,44);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,27,37,26,39,23,31)(21,33,43,45,35,46,29,42)(25,38,41,47,40,48,36,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,22,35)(18,25,26,40)(19,29,30,43)(20,33,34,46)(23,36,27,41)(24,38,39,48)(28,42,32,45)(31,44,37,47);(2,5)(6,8)(10,14)(13,15)(18,26)(20,34)(21,35)(23,27)(28,32)(29,43)(38,48)(44,47)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7237768);