Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130114 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2543 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,7,6,3,4,8)(9,10)(11,19)(12,23)(13,18,28,32)(14,29)(15,20)(16,22,33,26)(17,21)(24,31,27,40)(25,36)(30,35)(34,38,39,37);(1,3)(2,6)(4,8)(5,7)(9,11,15,14)(10,16,20,33)(12,21,25,30)(13,26,28,22)(17,31,35,40)(18,29,32,19)(23,39,36,34)(24,38,27,37);(1,4,6,5)(2,7,3,8)(9,12,15,25)(10,17,20,35)(11,21,14,30)(13,27,28,24)(16,31,33,40)(18,34,32,39)(19,36,29,23)(22,38,26,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,d^c=dg,e^c=ei,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37580);