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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2078 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)(7,9)(11,12)(13,14,17,16)(15,18,19,20)(21,22,26,27)(23,30,33,28)(24,25,35,29)(31,32,34,36);(5,6,10,8)(7,11,9,12)(15,19)(18,20)(22,27)(25,29)(28,30)(32,36);(5,7,10,9)(6,11,8,12)(13,15,17,19)(14,18,16,20)(21,23,26,33)(22,28,27,30)(24,34,35,31)(25,32,29,36);(1,2,3,4)(5,8,10,6)(7,12,9,11)(21,24)(22,29)(23,31)(25,27)(26,35)(28,32)(30,36)(33,34);(5,9,10,7)(6,12,8,11)(13,16,17,14)(15,20,19,18)(21,25,26,29)(22,24,27,35)(23,32,33,36)(28,34,30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=gh,d^2=gi,e^2=gh,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, 1106732);