Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500112 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
621 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,14,19,26,20)(11,15,23,36,24,34,21,32)(13,27,25,18,30,31,39,38)(16,29,37,22,35,40,33,28);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,23)(13,28)(14,24)(16,27)(17,36)(18,37)(19,34)(20,32)(21,26)(22,30)(25,29)(31,35)(33,38)(39,40);(9,11)(10,16)(12,21)(13,29)(14,24)(15,31)(17,33)(18,36)(19,35)(20,37)(22,39)(23,26)(25,28)(27,34)(30,40)(32,38);(2,6)(5,7)(9,12,14,26)(10,17,19,20)(11,21,24,23)(13,25,30,39)(15,32,34,36)(16,33,35,37)(18,31,38,27)(22,29,28,40)Polycyclic presentation:
2^9|a^2=gh,b^2=e,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265609, #72265615, #72265618, … |
| maximal normal subgroup of | #72265609, #72265615, #72265618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412781);