Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | |
1883 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)(15,26,30,21)(16,22,32,43)(17,33,20,23)(18,24,34,41)(27,44,35,37)(28,38,36,40)(29,42,31,39);(5,6)(7,10)(8,9)(11,12)(13,15,19,30)(14,21,25,26)(16,18,32,34)(17,28,20,36)(22,41,43,24)(23,38,33,40)(27,29,35,31)(37,42,44,39);(2,4)(13,16,17,29,19,32,20,31)(14,22,33,42,25,43,23,39)(15,27,36,34,30,35,28,18)(21,37,38,24,26,44,40,41);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,33)(15,28,30,36)(16,31,32,29)(18,27,34,35)(21,38,26,40)(22,42,43,39)(24,37,41,44);(6,9)(10,12)(13,18,17,27,19,34,20,35)(14,24,33,44,25,41,23,37)(15,29,36,32,30,31,28,16)(21,39,38,22,26,42,40,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715130);