Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3614 conjugacy classes of subgroups; only the 1010 normal subgroups are shown.
This group has 3614 conjugacy classes of subgroups — too many to draw. Only its 1010 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) = 512 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,7)(10,11)(13,14,15,16)(19,23)(22,24);(5,6,8,7)(18,21)(22,24);(9,10,12,11)(17,18)(19,22)(20,21)(23,24);(1,2,3,4)(5,7,8,6)(9,11,12,10)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61379);