Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94435 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1393 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | C4×#127 | C22 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C4×#127 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,15,13,17,22,16)(11,18,19,24,21,23,20,14);(2,5)(3,7)(9,11)(10,14,16,18)(12,20,22,19)(13,21)(15,23,17,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=fi,g^2=i,b^a=bc,c^a=cf,e^a=eghi,c^b=cg,d^b=dfhi,g^b=gh,h^b=hi,e^c=ehi,e^d=ei,g^e=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1673);