Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #195990 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C2×#2847 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2672 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)(11,15,16,13)(12,14)(17,18,20,24,21,22,19,23);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(18,22)(19,20);(2,6)(5,7)(17,19,21,20)(18,23,22,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=di,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72267822, #72267823 |
| maximal normal subgroup of | #72267822, #72267823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120701);