Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
10293 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,4)(6,8)(9,10,11,12);(2,4)(13,14,15,16);(9,10)(11,12)(13,14)(15,16);(2,4)(5,6)(7,8)(9,10)(11,12);(1,2)(3,4)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
| maximal subgroup of | #72213999, #72219693, #72231749, … |
| maximal normal subgroup of | #72213999, #72219693, #72231749, … |
| maximal quotient of | #72215961, #72233926, #72251013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 680976);