Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195972 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1388 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)(3,7)(4,5)(6,8)(9,10)(11,13,15,16)(12,14)(17,18,21,23,22,24,20,19);(2,5)(6,7)(9,11,12,15)(10,13)(14,16)(17,19,21,24,22,23,20,18);(1,3,4,8)(2,6,5,7)(10,14)(13,16)(17,20,22,21)(18,19,24,23);(17,20,22,21)(18,23,24,19)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=fh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,e^b=eh,g^b=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309178, #72309187, #72309242, … |
| maximal normal subgroup of | #72309178, #72309187, #72309242, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 114097);