Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C2×C4 | C82 |
| Frattini, cosocle | C2×C43 | C22 |
| socle, prime radical | C24 | #95 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C43 ▷ C24 ▷ C1 | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,19,21,20,22,23,24);(2,5,7,6)(9,11)(10,13,14,15)(12,16)(25,26,27,29,28,30,31,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=i,b^a=bc,c^a=cf,e^a=efg,c^b=cg152 conjugacy classes — view the full character table.
| maximal quotient of | #72237309, #72237310, #72237311 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10);