Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12099 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C22 | #1790 |
| prime radical | #392 | D4 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
608 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 608 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,15,12,17)(13,22,23,18)(14,19)(16,20)(21,24);(1,3)(2,6)(4,8)(5,7)(10,15)(11,12)(13,23)(16,22)(17,19)(18,24);(1,4,6,5)(2,7,3,8)(9,11,14,12)(10,16,19,24)(13,21,23,20)(15,18,17,22)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38763);