Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1682 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)(5,6,11,10)(7,8,12,9)(13,14)(15,20)(16,19)(17,18)(21,22)(23,29)(24,26)(25,28)(27,30)(31,36)(32,34)(33,35);(5,7)(6,9)(8,10)(11,12)(13,15,18,16)(14,19,17,20)(22,24)(23,30)(28,32)(31,33);(5,8,11,9)(6,12,10,7)(13,16,18,15)(14,19,17,20)(21,23)(22,27)(24,29)(25,33)(26,30)(28,36)(31,34)(32,35);(1,2)(3,4)(5,9,11,8)(6,12,10,7)(21,24,26,22)(23,29,30,27)(25,32,34,28)(31,36,33,35);(5,10,11,6)(7,8,12,9)(13,17,18,14)(15,20,16,19)(21,25,26,34)(22,28,24,32)(23,31,30,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4508130);