Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2974 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)(3,4)(6,8)(9,10,14,17,13,18,12,15)(11,19,22,23,20,16,21,24);(2,4)(10,15,18,17)(16,23,19,24);(5,6,7,8)(9,11,12,21,13,20,14,22)(10,16,15,23,18,19,17,24);(11,20)(16,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238300 |
| maximal normal subgroup of | #72238300 |
| maximal quotient of | #72238285 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261279);