Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1889 |
| prime radical | #18122 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 625 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 40 points:
(1,2,4,6,5,3,8,7)(9,10,14,19)(11,20,24,38)(12,25,21,17)(13,18)(15,30,33,22)(16,28)(23,37)(26,35)(27,36)(29,34)(31,40)(32,39);(1,3)(2,5)(4,6)(7,8)(9,11,21,30,14,24,12,22)(10,15,25,38,19,33,17,20)(13,27,29,39,28,23,26,40)(16,31,34,37,18,32,35,36);(1,2)(3,5)(4,6)(7,8)(10,16)(13,28)(15,31)(17,35)(18,19)(20,36)(23,27)(25,34)(26,29)(32,33)(37,38)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=gi,f^2=g,b^a=bd,c^a=ce,d^a=di,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44563);