Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484583 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #23443 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1037 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1037 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(2,6)(3,4)(7,8)(9,10)(11,20)(12,18)(13,17)(14,19)(15,21)(16,25)(22,35)(23,28)(24,33)(26,40)(27,34)(29,36)(30,37)(31,39)(32,38);(1,2)(3,6)(4,8)(5,7)(9,11,14,23)(10,15,19,30)(12,21,25,37)(13,26,27,22)(16,28,18,20)(17,32,34,29)(24,38,39,36)(31,35,33,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16,19,18)(11,21)(13,24)(14,25)(15,28,30,20)(17,31,34,33)(22,36)(23,37)(26,38)(27,39)(29,40,32,35);(9,13,14,27)(10,17,19,34)(11,22,23,26)(12,24,25,39)(15,29,30,32)(16,31,18,33)(20,35,28,40)(21,36,37,38)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391971);