Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480454 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35129 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14,21,24)(12,17)(13,16)(15,18)(19,23,20,22);(9,11,14,18,15,21,24,10)(12,19,22,17,13,20,23,16);(2,5)(6,8)(9,12,15,13)(10,16,18,17)(11,19,21,20)(14,22,24,23);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,17,18,16)(11,20,21,19)(14,23,24,22)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=i,d^2=f,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311263, #72311264, #72311273, … |
| maximal normal subgroup of | #72311263, #72311264, #72311273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402779);