Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(17,18,21,22,20,24,19,23);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16);(9,11,12,16)(10,13,14,15)(18,22)(19,21)(23,24);(2,6)(5,7);(3,8)(5,7)(10,14)(13,15)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72283458, #72305299, #72305302, … |
| maximal normal subgroup of | #72283458, #72305299, #72305302, … |
| maximal quotient of | #72305656, #72305809, #72316445 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6664546);