Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #298661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
880 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,14,12,16,11,15)(18,19)(22,24);(1,3,5,4)(2,6,8,7)(17,18,21,22,20,19,23,24);(10,14)(11,13)(15,16)(17,19)(18,20)(21,22)(23,24);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^9|a^2=efh,b^2=egi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72224775, #72224777, #72224783 |
| maximal normal subgroup of | #72224775, #72224777, #72224783 |
| maximal quotient of | #72226806 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 230644);