Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 | |
9186 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)(5,6)(7,8)(9,10,11,12);(5,6,7,8)(14,16);(1,2)(3,4)(9,10)(11,12)(13,14)(15,16);(5,6,7,8)(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213719, #72219266, #72219499, … |
| maximal normal subgroup of | #72213719, #72219261, #72219266, … |
| maximal quotient of | #72213603, #72250287, #72251798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1325589);