Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1867 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(2,6)(3,4)(7,8)(9,10,13,18,14,19,26,34)(11,20,23,35,22,28,37,39)(12,17,25,33,24,16,31,32)(15,27,29,36,30,38,40,21);(1,2)(3,6)(4,8)(5,7)(9,11,13,23,14,22,26,37)(10,15,18,29,19,30,34,40)(12,21,25,27,24,36,31,38)(16,28,32,39,17,20,33,35);(1,3,5,4)(2,6,7,8)(9,12)(10,16,19,17)(11,21)(13,25)(14,24)(15,28,30,20)(18,32,34,33)(22,36)(23,27)(26,31)(29,39,40,35)(37,38)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gh,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dg,e^b=ehi,d^c=dh,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43276);