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 | |
2274 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)(5,6,9,12)(7,10,11,8)(13,14,16,18)(15,19,20,17)(21,22)(23,26)(24,27)(25,30)(28,34)(29,32)(31,36)(33,35);(5,7,9,11)(6,10,12,8)(13,14,16,18)(15,17,20,19)(22,23)(27,29)(28,30)(35,36);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35);(5,8,9,10)(6,11,12,7)(13,15)(14,17)(16,20)(18,19)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,20)(18,19)(21,25)(22,28)(23,30)(24,31)(26,34)(27,35)(29,36)(32,33)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=hi,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, 1220113);