Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500891 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
520 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 144 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)(4,5)(6,8)(9,10)(11,16)(12,17)(13,19)(14,28)(15,20)(18,26)(21,35)(22,30)(23,32)(24,31)(25,33)(27,34)(29,36)(37,40)(38,39);(2,5)(6,7)(9,11,14,21,15,22,13,24)(10,16,28,35,20,30,19,31)(12,23,26,36,27,37,25,38)(17,32,18,29,34,40,33,39);(10,17,20,34)(11,21)(12,25)(13,14)(16,29,30,39)(18,28,33,19)(22,24)(26,27)(31,40,35,32)(36,38);(1,3,4,8)(2,6,5,7)(11,22)(16,30)(21,24)(23,37)(29,39)(31,35)(32,40)(36,38)Polycyclic presentation:
2^9|b^2=gi,c^2=gh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238779, #72239796, #72239832, … |
| maximal normal subgroup of | #72238779, #72239796, #72239832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 414771);