Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139712 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
1236 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 1236 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,15,21,22)(11,17,19,24,18,14,20,23);(1,2)(3,5)(4,6)(7,8)(9,11,12,20)(10,14,22,24)(13,18,21,19)(15,17,16,23);(1,3)(2,5)(4,8)(6,7)Polycyclic presentation:
2^9|a^2=f,b^2=df,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72233159, #72233165, #72233166, … |
| maximal normal subgroup of | #72233159, #72233165, #72233166, … |
| maximal quotient of | #72266325 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46910);