Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#3184 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1034 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(2,5,6,7)(9,10,13,16)(11,18)(12,15,22,17)(14,21)(19,24)(20,23);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,19)(13,21)(15,23)(16,18)(17,24)(20,22);(1,2)(3,7)(4,6)(5,8)(10,15)(11,19)(12,22)(16,17)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219656 |
| maximal normal subgroup of | #72219656 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18586);