Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5)(3,7)(6,8)(9,10,11,14,12,13,15,16);(1,2)(3,8)(4,6)(5,7);(3,7)(5,8);(10,13)(14,16)Polycyclic presentation:
2^9|a^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,f^b=fh,e^c=eh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72226690, #72239485, #72278460, … |
| maximal normal subgroup of | #72226690, #72239485, #72278460, … |
| maximal quotient of | #72219603 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92135);