Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133922 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2308 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
895 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 82 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)(12,21,23,16)(13,17)(14,19,24,22)(15,20);(9,11)(10,14)(12,19)(13,20)(15,21)(16,18)(17,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,18)(12,22)(13,20)(14,21,24,16)(19,23)Polycyclic presentation:
2^9|a^2=f,c^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fh,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72255651, #72255673, #72255704, … |
| maximal normal subgroup of | #72255651, #72255673, #72255704, … |
| maximal quotient of | #72263133 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41146);