Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34636 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1859 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)(13,14,17,22,18,21,29,36)(15,19,24,32,25,31,39,43)(16,20,28,35,27,34,37,41)(23,30,38,42,26,33,40,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,24,18,25,29,39)(14,19,22,32,21,31,36,43)(16,26,28,40,27,23,37,38)(20,33,35,44,34,30,41,42);(2,4)(13,16)(14,20)(15,23)(17,28)(18,27)(19,30)(21,34)(22,35)(24,38)(25,26)(29,37)(31,33)(32,42)(36,41)(39,40)(43,44);(6,9)(10,12)(14,21)(16,27)(19,31)(22,36)(23,26)(28,37)(32,43)(38,40);(5,7,8,11)(6,10,9,12)(16,27)(20,34)(23,26)(28,37)(30,33)(35,41)(38,40)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6295581);