Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 | #52083 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1051 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 134 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)(11,21)(12,14)(13,26)(16,30)(17,19)(18,33)(22,37)(23,39)(24,27)(28,38)(31,34)(35,40);(2,5)(6,8)(9,10,15,20)(11,22,24,35)(12,19,14,17)(13,18,26,33)(16,28,31,23)(21,37,27,40)(25,36,29,32)(30,38,34,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,23)(13,27)(14,28)(15,24)(17,22)(18,34)(19,35)(20,31)(21,26)(25,39)(29,38)(30,33)(32,37)(36,40);(2,5)(6,8)(9,12,15,14)(10,17,20,19)(11,23,24,28)(13,25,26,29)(16,22,31,35)(18,32,33,36)(21,38,27,39)(30,40,34,37)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=dei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407412);