Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(14,18)(16,20)(21,22,26,30)(23,27,28,24)(25,29,34,36)(31,33,35,32);(2,4)(14,18)(15,19)(21,23,26,28)(22,24,30,27)(25,31,34,35)(29,32,36,33);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,16,19,20)(21,23)(22,27)(24,30)(25,31)(26,28)(29,33)(32,36)(34,35);(5,7)(6,10)(8,12)(9,11)(13,15,17,19)(14,16,18,20)(21,24,26,27)(22,28,30,23)(25,33,34,32)(29,31,36,35);(5,8,9,6)(7,12,11,10)(13,16)(14,19)(15,18)(17,20)(21,25,26,34)(22,29,30,36)(23,31,28,35)(24,32,27,33)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2661644);