Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | C23×C22⋊C4 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(11,18,20,22)(15,23,24,19);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,17)(16,21)(18,23)(19,22)(20,24);(9,11,13,22,14,20,16,18)(10,15,12,19,17,24,21,23);(3,8)(5,7)(9,12)(10,16)(11,19)(13,17)(14,21)(15,18)(20,23)(22,24);(2,6)(5,7)(10,17)(12,21)(15,24)(19,23)Polycyclic presentation:
2^9|a^2=h,c^2=gh,g^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300569);