Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,7,8)(13,14,17,20)(15,19,16,18);(1,2,4,3)(9,10,11,12)(15,16)(18,19);(2,3)(9,10)(11,12)(13,15,17,16)(14,18,20,19);(5,6)(7,8)(13,16,17,15)(14,19,20,18);(1,3,4,2)(13,14)(15,18)(16,19)(17,20)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72278988 |
| maximal normal subgroup of | #72278988 |
| maximal quotient of | #72279137, #72299049, #72299694 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2017991);