Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | |
2147 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(7,11)(10,12)(13,14,16,18)(15,19,20,17)(21,22,26,27)(23,31,32,28)(24,34,35,29)(25,33,36,30);(5,6,9,8)(7,10,11,12)(13,14,16,18)(15,17,20,19)(22,27)(23,32)(29,34)(30,33);(1,2,4,3)(13,14)(15,17)(16,18)(19,20)(21,23,26,32)(22,28,27,31)(24,33,35,30)(25,29,36,34);(5,7)(6,10)(8,12)(9,11)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,28,36,31);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,15)(14,17)(16,20)(18,19)(21,25,26,36)(22,30,27,33)(23,29,32,34)(24,31,35,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=fgh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1275289);