Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130062 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #46384 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
851 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 851 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,12,16,13,15,20,23)(11,17,19,24,18,14,21,22);(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,24)(22,23);(10,15)(11,18)(12,20)(14,17)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fi,e^d=ei,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72213224, #72213776, #72213786, … |
| maximal normal subgroup of | #72213224, #72213776, #72213786, … |
| maximal quotient of | C2×#130068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37264);