Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393909 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(5,6)(7,8)(17,18,19,23,21,24,20,22);(2,4)(9,10,12,14)(11,15,16,13)(18,22)(19,20)(23,24);(6,8)(9,11,12,16)(10,13,14,15);(18,23)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=g,c^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72286750 |
| maximal normal subgroup of | #72286750 |
| maximal quotient of | #72287672 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310575);