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 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,12)(7,10,11,8)(14,17)(15,19)(21,22,26,27)(23,29,30,24)(25,33,34,28)(31,32,36,35);(1,2,3,4)(5,7,9,11)(6,10,12,8)(14,17)(18,20)(22,27)(23,30)(25,34)(32,35);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,16,17)(15,18,19,20)(21,23,26,30)(22,24,27,29)(25,31,34,36)(28,32,33,35);(5,6)(7,10)(8,11)(9,12)(21,24)(22,23)(25,32)(26,29)(27,30)(28,31)(33,36)(34,35);(5,8,9,10)(6,11,12,7)(13,15,16,19)(14,18,17,20)(21,25,26,34)(22,28,27,33)(23,31,30,36)(24,32,29,35)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=gi,e^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, 1077054);