Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,11,12)(7,8,10,9)(14,16)(15,19)(21,22)(23,29)(24,28)(25,26)(27,30)(31,35)(32,33)(34,36);(5,7,11,10)(6,9,12,8)(13,14,17,16)(15,18,19,20)(22,26)(23,30)(28,32)(31,36);(5,8,11,9)(6,7,12,10)(13,15)(14,18)(16,20)(17,19)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,35)(34,36);(1,2)(3,4)(5,9,11,8)(6,7,12,10)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35);(5,10,11,7)(6,8,12,9)(13,16,17,14)(15,18,19,20)(21,24,25,33)(22,28,26,32)(23,31,30,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fgh,d^2=h,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1996750);