Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1147 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)(3,7)(4,5)(6,8)(9,10)(11,13,18,22)(12,19)(14,16)(15,20)(17,23,24,21);(2,5)(6,7)(9,11,13,16,14,18,22,10)(12,17,21,19,20,24,23,15);(1,3,4,8)(2,6,5,7)(9,12,14,20)(10,15,16,19)(11,17,18,24)(13,21,22,23);(12,20)(15,19)(17,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=f,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311476, #72311478, #72311510, … |
| maximal normal subgroup of | #72311476, #72311478, #72311510, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402596);