Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C23 | C2×#135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,19,15,21,30,20)(11,22,27,37,23,35,40,38)(12,17,13,29,24,18,31,36)(16,25,33,39,32,28,34,26);(1,3,5,4)(2,6,7,8)(11,23)(16,32)(22,35)(25,28)(26,39)(27,40)(33,34)(37,38);(9,11,24,25,14,27,31,39,15,23,12,28,30,40,13,26)(10,16,29,38,20,34,17,35,21,32,36,37,19,33,18,22);(1,4,5,3)(2,6,7,8)(9,12,14,13,15,24,30,31)(10,17,19,29,21,18,20,36)(11,25,27,39,23,28,40,26)(16,22,33,37,32,35,34,38)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fh,d^2=eg,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72210514 |
| maximal normal subgroup of | #72210514 |
| maximal quotient of | #72210412 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403361);