Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,11)(7,10,12,9)(13,14)(15,17)(16,18)(19,20)(21,22,25,27)(23,28,31,34)(24,32,26,29)(30,36,33,35);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(22,27)(23,30)(24,26)(28,35)(31,33)(34,36);(5,7)(6,10)(8,12)(9,11)(14,18)(17,20)(21,23,24,33,25,31,26,30)(22,28,32,35,27,34,29,36);(7,12)(9,10)(15,19)(17,20)(21,24,25,26)(22,29,27,32)(23,30,31,33)(28,35,34,36)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=gi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299246);