Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×2−1+4 | C2×C4 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
516 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 516 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,7,4,3,6,8)(9,10,14,19)(11,20)(12,17,21,30)(13,18)(15,25)(16,28)(22,33)(23,34,27,31)(24,37)(26,36)(29,35)(32,38,40,39);(1,3,5,8,4,2,6,7)(9,11,21,22,14,24,12,25)(10,15,30,20,19,33,17,37)(13,27,28,39,29,23,26,38)(16,34,35,40,36,31,18,32);(2,3)(7,8)(9,12,14,21)(10,16,19,36)(11,22,24,25)(13,28,29,26)(15,31,33,34)(17,18,30,35)(20,32,37,40)(23,39,27,38)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=egi,g^a=gi,c^b=chi,d^b=di,e^b=egi,f^b=fi,d^c=dg,e^c=ei,f^c=fg,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, 44052);