Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7563341 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,22)(13,19)(14,18)(15,17)(16,21)(23,24);(2,5)(6,8)(9,11)(10,16)(12,13)(14,21)(15,23)(17,22)(18,20)(19,24);(1,3,4,7)(2,6,5,8)(9,12,13,11,14,23,15,21)(10,16,17,24,18,20,19,22);(10,17,18,19)(16,24,20,22);(2,5)(6,8)(9,13,14,15)(10,17,18,19)(11,12,21,23)(16,22,20,24)Polycyclic presentation:
2^9|c^2=fg,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217796 |
| maximal normal subgroup of | #72217796 |
| maximal quotient of | #72216146, #72216148, #72216163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7520672);