Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511802 |
| 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 | C22×Q8 | C24 |
| prime radical | #23909 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,25)(13,20)(14,29)(15,31)(16,21)(18,27)(19,23)(22,38)(24,36)(26,34)(28,37)(30,35)(32,39)(33,40);(2,5)(6,8)(9,11,16,15)(10,17,21,31)(12,26,28,22)(13,23,14,24)(18,33,35,32)(19,29,36,20)(25,34,37,38)(27,40,30,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,27)(14,30)(15,26)(16,28)(17,32)(19,34)(20,37)(21,35)(23,39)(24,40)(25,29)(31,33)(36,38);(9,13,16,14)(10,19,21,36)(11,23,15,24)(12,27,28,30)(17,29,31,20)(18,34,35,38)(22,39,26,40)(25,33,37,32)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419331);