Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511779 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#120 | C23 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(1,3,4,8)(2,5,6,7)(11,18)(14,21)(17,23)(19,24);(9,11,13,18)(10,14,16,21)(12,19,20,24)(15,17,22,23);(12,20)(19,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,g^c=ghi,f^d=fh,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72281983, #72282019, #72282042 |
| maximal normal subgroup of | #72281983, #72282019, #72282042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419228);