Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133924 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #34743 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
1015 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 1015 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10)(11,18,19,14)(12,16,21,15)(13,17)(20,23)(22,24);(1,3,4,8)(2,5,6,7)(10,14,17,18)(11,19)(12,21)(15,22,16,23);(10,15)(11,19)(12,21)(14,22)(16,17)(18,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=di,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72238365, #72262728, #72262731 |
| maximal normal subgroup of | #72238365, #72262728, #72262731 |
| maximal quotient of | #72238366 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41120);