Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93514 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C23×C22⋊C4 | C22 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C22⋊C4 ▷ C24 ▷ C2 ▷ C1 | |
1022 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,20,22,15)(13,16)(14,18,23,21)(19,24);(2,5,7,6)(9,11)(10,14)(12,21,22,18)(13,19)(15,24,20,17)(16,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,b^a=bc,c^a=cf,e^a=efgh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227893, #72227898, #72237040 |
| maximal normal subgroup of | #72227893, #72227898, #72237040 |
| maximal quotient of | #72238708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 710);