Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508078 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #23915 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1104 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 124 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)(9,10,15,20)(11,21,24,34)(12,19,14,17)(13,26,28,18)(16,27,32,23)(22,37,29,38)(25,33,30,36)(31,39,35,40);(2,5)(6,8)(11,22)(12,14)(16,31)(17,19)(21,37)(23,40)(24,29)(25,30)(27,39)(32,35)(33,36)(34,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,27)(14,29)(15,24)(17,31)(18,34)(19,35)(20,32)(21,26)(23,28)(25,40)(30,39)(33,37)(36,38);(3,7)(6,8)(9,12,15,14)(10,17,20,19)(11,22,24,29)(13,25,28,30)(16,31,32,35)(18,33,26,36)(21,38,34,37)(23,39,27,40)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 415280);