Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #38023 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#546 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23)(18,21,24,22);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,14,16,23)(12,18,20,24)(15,22,19,21);(1,3,4,7)(2,6,5,8)(10,15)(11,17)(14,21)(16,19)(18,24)(22,23);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=e,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285728, #72285788, #72286181, … |
| maximal normal subgroup of | #72285728, #72285788, #72286181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404714);