Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393877 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#537 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10)(11,18,21,23)(12,16)(13,17)(14,19,15,20)(22,24);(1,2)(3,4)(9,11)(10,14)(12,20)(13,21)(15,17)(16,23)(18,24)(19,22);(2,4)(9,11)(10,15)(12,19)(13,21)(14,17)(16,23)(18,24)(20,22);(6,8)Polycyclic presentation:
2^9|a^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72264462, #72264531, #72264535 |
| maximal normal subgroup of | #72264462, #72264531, #72264535 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302582);