Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213564 |
| 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, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(17,18)(19,23,24,21)(20,22);(1,2,4,5)(3,8,7,6)(9,10,12,14)(11,15,16,13);(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,24)(22,23);(1,3,4,7)(2,6,5,8)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72227402, #72227407 |
| maximal normal subgroup of | #72227402, #72227407 |
| maximal quotient of | #72267697, #72268081, #72268110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294296);