Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,13,18,14,17,23,24)(11,15,20,21,19,12,16,22);(1,2,4,3)(5,7,8,6)(9,11)(10,15)(12,17)(13,20)(14,19)(16,23)(18,21)(22,24);(1,3,4,2)(6,7)(9,12)(10,16)(11,18)(13,22)(14,15)(17,20)(19,24)(21,23);(10,17)(11,19)(16,20)(18,24)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238811 |
| maximal normal subgroup of | #72238811 |
| maximal quotient of | #72238820, #72240017 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265425);