Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1114 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 1114 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:
(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(21,22)(23,27)(24,29)(25,30)(26,28)(31,34)(32,36)(33,35);(1,2)(3,4)(6,9)(7,11)(14,17)(18,19)(21,23,25,31,26,33,24,32)(22,27,30,34,28,35,29,36);(2,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(23,31)(24,25)(27,34)(29,30)(32,33)(35,36);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(22,28)(27,35)(29,30)(34,36)Polycyclic presentation:
2^9|a^2=e,b^2=h,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,d^c=de,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299135);