Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132541 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
529 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,20,23,33)(12,24,27,31)(13,18,28,17)(14,19)(15,22)(16,26)(21,29)(25,32)(30,38)(34,35,39,36)(37,40);(1,2)(3,7)(4,6)(5,8)(9,11,14,23)(10,15)(12,21,27,37)(13,22,28,38)(16,29)(17,20)(18,33)(19,30)(24,39)(25,35,26,36)(31,34)(32,40);(3,8)(5,7)(9,12,14,27)(10,16,19,32)(11,21,23,37)(13,26,28,25)(15,29,30,40)(17,31,18,24)(20,34,33,39)(22,36,38,35)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=i,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=egh,f^b=fghi,d^c=dghi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39924);