Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480690 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1822 conjugacy classes of subgroups — too many to draw. Only its 206 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)(11,19,22,34)(12,16)(13,26,28,17)(14,18)(15,21)(20,35,24,38)(23,31)(25,40,37,33)(27,30)(29,36)(32,39);(1,3)(2,5)(4,8)(6,7)(12,23)(16,31)(20,24)(25,37)(29,32)(33,40)(35,38)(36,39);(3,8)(5,7)(9,11,14,22)(10,15,18,30)(12,24,23,20)(13,27,28,21)(16,32,31,29)(17,34,26,19)(25,36,37,39)(33,35,40,38);(2,6)(5,7)(9,12,14,23)(10,16,18,31)(11,20,22,24)(13,25,28,37)(15,29,30,32)(17,33,26,40)(19,35,34,38)(21,36,27,39)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72310930, #72311055, #72311064, … |
| maximal normal subgroup of | #72310930, #72311055, #72311064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403401);