Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132529 |
| 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 | #52762 | C2 |
| 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 | |
597 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 597 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10)(11,20)(12,24)(13,27,14,18)(15,19)(16,22,31,28)(17,25,32,29)(21,30)(23,35)(26,33)(34,37,39,36)(38,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,30)(18,20)(19,31)(24,39)(25,36)(26,38)(27,35)(29,37)(32,40)(33,34);(3,8)(5,7)(9,12,15,26)(10,17,19,32)(11,21,23,38)(13,25,14,29)(16,30,31,40)(18,33,27,24)(20,34,35,39)(22,36,28,37)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fg,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, 39726);