Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | #1867 |
| prime radical | C22×D4 | C4⋊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 | |
347 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 347 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,3,4,6,8,7)(9,10)(11,14)(12,15)(13,18)(16,19)(17,21)(20,23)(22,24);(1,3,5,6,4,7,8,2)(10,14)(11,19)(12,21)(15,23)(16,18)(17,24);(2,6)(3,7)(10,15)(12,21)(14,23)(16,24)(17,18)(20,22)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=egi,d^c=dg,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43158);