Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8346 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#534 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(10,13,14,15)(17,18,19,21,20,22,23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16);(1,2)(3,5)(4,6)(7,8)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219584, #72219603, #72219625 |
| maximal normal subgroup of | #72219584, #72219603, #72219625 |
| maximal quotient of | #72219576, #72219595, #72219596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4134);