Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470225 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(3,7)(6,8)(9,10)(11,15,16,13)(12,14);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238110, #72238193, #72251274, … |
| maximal normal subgroup of | #72238110, #72238193, #72251274, … |
| maximal quotient of | #72257666, #72316521, C2×#6491478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398674);