Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352157 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
4023 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)(3,5)(4,6)(7,8)(14,20)(21,24);(2,5)(3,8)(6,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(9,11,13,18)(10,14,16,20)(12,17,19,23)(15,21,22,24);(2,6)(5,7)(9,12)(10,15)(11,17)(13,19)(14,21)(16,22)(18,23)(20,24);(12,19)(15,22)(17,23)(21,24)Polycyclic presentation:
2^9|c^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281484, #72281528, #72281920 |
| maximal normal subgroup of | #72281484, #72281528, #72281920 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259362);