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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,8)(10,11)(14,18)(15,16)(21,22,26,29)(23,30,24,27)(25,28,33,36)(31,35,32,34);(2,4)(21,23,26,24)(22,27,29,30)(25,32,33,31)(28,35,36,34);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,24,26,23)(22,27,29,30)(25,32,33,31)(28,34,36,35);(5,7,9,12)(6,10,8,11)(13,15,17,16)(14,19,18,20)(21,22)(23,27)(24,30)(25,28)(26,29)(31,34)(32,35)(33,36);(5,8,9,6)(7,11,12,10)(13,16,17,15)(14,20,18,19)(21,25,26,33)(22,28,29,36)(23,31,24,32)(27,34,30,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=g,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, 1049848);