Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
950 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,22)(13,24)(14,17)(15,18)(16,20)(21,23);(2,5)(6,7)(11,16)(12,23)(19,20)(21,22);(9,11,20,17,13,21,23,24,14,16,19,10,15,22,12,18);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72224853 |
| maximal normal subgroup of | #72224853 |
| maximal quotient of | #72224844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410066);