Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | #62815 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3411 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)(9,10,13,17,14,12,21,16)(11,15,20,23,19,18,24,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(1,3,4,7)(2,6,5,8)(9,12)(10,14)(11,18)(13,16)(15,19)(17,21)(20,22)(23,24);(2,5)(6,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,22)(17,23)(21,24);(11,19)(15,18)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,g^c=gi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72303398 |
| maximal normal subgroup of | #72303398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286535);