Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394747 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 214 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:
(1,2,5,6)(3,7,4,8)(9,10,15,18,14,19,13,29)(11,20,23,33,24,32,22,16)(12,25,30,34,27,38,28,17)(21,31,37,35,26,39,36,40);(2,6)(7,8)(9,11,13,22,14,24,15,23)(10,16,18,20,19,33,29,32)(12,26,28,37,27,21,30,36)(17,31,25,35,34,39,38,40);(1,3,5,4)(2,7,6,8)(9,12,13,28,14,27,15,30)(10,17,18,25,19,34,29,38)(11,21,22,36,24,26,23,37)(16,31,20,35,33,39,32,40);(1,4,5,3)(2,7,6,8)(12,27)(17,34)(21,26)(25,38)(28,30)(31,39)(35,40)(36,37)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=e,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324061);