Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2550 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)(5,6,7,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24);(2,4)(10,15,18,17)(16,22,23,24);(9,11,13,21,14,19,12,20)(10,16,17,24,18,23,15,22);(11,19)(16,23)(20,21)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,e^2=i,h^2=i,b^a=be,e^a=ei,g^a=gi,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72238842, #72239938 |
| maximal normal subgroup of | #72238842, #72239938 |
| maximal quotient of | #72238846 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260341);