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 | |
2038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,7)(8,12)(15,20)(17,19)(21,22,25,26)(23,29,30,27)(24,28,33,34)(31,36,32,35);(2,4)(13,14,18,16)(15,19,20,17)(22,26)(23,30)(28,34)(31,32);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,15,18,20)(14,19,16,17)(21,23,25,30)(22,27,26,29)(24,32,33,31)(28,36,34,35);(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,17,20,19)(21,24,25,33)(22,28,26,34)(23,31,30,32)(27,35,29,36);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,36)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=g,d^2=g,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, 1021872);