Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15,16,13)(12,14)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8);(9,11)(10,13)(12,16)(14,15)(18,21)(19,24)(22,23);(1,3,4,7)(2,6,5,8)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264297, #72308033, #72308052, … |
| maximal normal subgroup of | #72264297, #72308033, #72308052, … |
| maximal quotient of | C2×#191038 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98234);