Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,25)(15,26,29,21)(16,30,27,22)(17,32,20,23)(18,34,36,24)(28,38,43,42)(31,39,35,41)(33,40,37,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,21,25,26)(16,17,27,20)(18,31,36,35)(22,32,30,23)(24,39,34,41)(28,37,43,33)(38,40,42,44);(2,4)(13,16,20,15,19,27,17,29)(14,22,23,21,25,30,32,26)(18,35,37,43,36,31,33,28)(24,41,40,42,34,39,44,38);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,32)(15,27,29,16)(18,33,36,37)(21,22,26,30)(24,40,34,44)(28,35,43,31)(38,39,42,41);(6,9)(10,12)(13,18,17,33,19,36,20,37)(14,24,32,44,25,34,23,40)(15,28,16,31,29,43,27,35)(21,38,22,39,26,42,30,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715194);