Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1610 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)(3,7,8,5)(4,6)(9,10,12,14,13,15,11,16)(17,18,20,23,21,24,19,22);(1,3)(2,5)(4,8)(6,7)(10,14)(11,12)(15,16);(10,15)(14,16)(17,19,21,20)(18,22,24,23);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,c^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,d^a=dg,e^a=eg,e^b=eh,f^b=fgh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72309120, #72309121, #72309125, … |
| maximal normal subgroup of | #72309120, #72309121, #72309125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67787);