Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184792 |
| 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 | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2746 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)(4,6)(5,8)(9,10)(11,15,16,14)(12,13)(17,18,20,22,19,21,23,24);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(18,21)(22,24);(3,8)(5,7)(9,11)(10,14)(12,16)(13,15)(18,21)(20,23);(2,6)(5,7)(10,13)(14,15)Polycyclic presentation:
2^9|a^2=g,b^2=eh,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72228190, #72228194, #72228196 |
| maximal normal subgroup of | #72228190, #72228194, #72228196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122325);