Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301973 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1276 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 152 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,21)(13,20)(14,22)(15,19)(16,18)(23,24);(1,3)(2,5)(4,8)(6,7)(9,11,14,20,15,12,16,23)(10,13,18,17,19,24,22,21);(1,2)(3,7,8,5)(4,6)(9,12,15,11)(10,17,19,21)(13,18,24,22)(14,23,16,20);(9,13,14,17,15,24,16,21)(10,11,18,20,19,12,22,23)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,g^c=ghi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209169);