Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
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 | C2×#2657 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 1892 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)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,35)(31,34)(32,36);(1,2,3,4)(6,9)(7,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,26)(24,29)(25,31)(27,34)(28,33)(30,32)(35,36);(7,11)(10,12)(14,18)(17,20)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(5,7,8,11)(6,10,9,12)(15,19)(17,20)(22,27)(26,34)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299056);