Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10)(11,17)(12,20,21,15)(13,16)(14,19)(18,22,24,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(2,5)(6,8)(11,19)(14,17)(18,24)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,e^2=fi,b^a=bf,c^a=cg,g^a=gi,d^b=df,h^c=hi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72268288, #72268293, #72268316, … |
| maximal normal subgroup of | #72268288, #72268293, #72268316, … |
| maximal quotient of | C4×D4×#99, #72313317 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286701);