Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,22,17)(11,18,20,24,19,14,21,23);(2,5)(3,7)(9,11)(10,14)(12,21)(13,19)(15,24)(16,18)(17,23)(20,22);(14,18)(23,24);(1,3,4,7)(2,6,5,8)(10,15,16,17)(14,23,18,24)Polycyclic presentation:
2^9|a^2=gh,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72318058, #72318294, #72318340 |
| maximal normal subgroup of | #72318058, #72318294, #72318340 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358218);