Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60900 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#432 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,18,15,19,13,27)(11,16,23,31,21,30,22,37)(12,24,25,17,26,32,28,33)(20,34,36,29,35,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,23,15,21,13,22)(10,16,18,31,19,30,27,37)(12,20,25,36,26,35,28,38)(17,29,32,40,33,39,24,34);(9,12,14,25,15,26,13,28)(10,17,18,32,19,33,27,24)(11,20,23,36,21,35,22,38)(16,29,31,40,30,39,37,34);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(20,35)(22,23)(29,39)(31,37)(34,40)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6302512);