Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500010 |
| 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 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(9,10)(11,15,16,13)(12,14);(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15);(3,8)(5,7);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72210578, #72213349, #72213719, … |
| maximal normal subgroup of | #72210578, #72213349, #72213719, … |
| maximal quotient of | #72210487, #72213210, #72213335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407206);