Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1436 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1436 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,4)(6,7)(9,10)(11,21)(12,18)(13,29)(14,19)(15,20)(16,31)(17,23)(22,34)(24,38)(25,40)(26,33)(27,35)(28,36)(30,37)(32,39);(5,6,8,7)(9,11,15,24,14,23,16,13)(10,17,20,29,19,21,31,38)(12,25,28,39,27,26,30,22)(18,33,36,34,35,40,37,32);(2,4)(6,7)(9,12)(10,18)(11,22)(13,25)(14,27)(15,30)(16,28)(17,32)(19,35)(20,37)(21,34)(23,39)(24,26)(29,40)(31,36)(33,38);(5,7,8,6)(9,13,16,23,14,24,15,11)(10,17,20,29,19,21,31,38)(12,26,28,22,27,25,30,39)(18,34,37,33,35,32,36,40)Polycyclic presentation:
2^9|b^2=g,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 382935);