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 | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
677 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 677 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)(9,10,13,28,14,18,29,19)(11,15,22,40,20,31,37,33)(12,23,26,17,27,16,25,36)(21,34,39,32,24,30,38,35);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,24)(13,22)(14,20)(16,34)(17,35)(18,31)(19,33)(21,27)(23,30)(25,39)(26,38)(28,40)(29,37)(32,36);(10,16)(12,25)(13,29)(15,30)(17,28)(18,23)(19,36)(21,38)(22,37)(24,39)(26,27)(31,34)(32,40)(33,35);(2,6)(5,7)(11,20)(15,31)(21,24)(22,37)(30,34)(32,35)(33,40)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=e,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 | #72265226, #72265461, #72265609, … |
| maximal normal subgroup of | #72265226, #72265461, #72265609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412779);