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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2014 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 2014 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)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,34)(31,35)(32,36);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(23,29)(24,31)(26,33)(28,35)(30,32)(34,36);(5,7)(6,10)(8,12)(9,11)(14,18)(17,20)(21,23,25,30)(22,26,27,34)(24,32,31,29)(28,36,35,33);(7,12)(9,10)(15,19)(17,20)(22,27)(26,34)(28,35)(33,36)Polycyclic presentation:
2^9|b^2=h,c^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=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298901);