Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(10,14,15,16);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15);(1,3)(2,5)(4,8)(6,7);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,g^b=ghi,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72258226, #72267299, #72267442, … |
| maximal normal subgroup of | #72258226, #72267299, #72267442, … |
| maximal quotient of | #72258433, #72267255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92653);