Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,12)(7,8,11,9)(14,17)(15,19)(21,22)(23,28)(24,30)(25,29)(26,27)(31,35)(32,36)(33,34);(5,7,10,11)(6,8,12,9)(14,17)(18,20)(22,27)(23,30)(25,33)(32,35);(5,8,10,9)(6,7,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,30)(22,28,27,24)(25,31,33,36)(29,35,34,32);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(21,24,26,28)(22,23,27,30)(25,32,33,35)(29,31,34,36);(5,9,10,8)(6,11,12,7)(13,15,16,19)(14,18,17,20)(21,25,26,33)(22,29,27,34)(23,31,30,36)(24,32,28,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 994036);