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 | |
1768 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)(13,14,17,20)(15,16,19,18)(22,23)(24,31)(25,33)(35,36);(2,4)(15,19)(16,18)(21,22,26,23)(24,27,31,29)(25,28,33,30)(32,35,34,36);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15,17,19)(14,18,20,16)(21,23,26,22)(24,27,31,29)(25,30,33,28)(32,35,34,36);(5,7)(6,10)(8,12)(9,11)(13,16,17,18)(14,19,20,15)(21,24)(22,27)(23,29)(25,34)(26,31)(28,36)(30,35)(32,33);(5,8,9,6)(7,12,11,10)(13,14)(15,18)(16,19)(17,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgi,d^2=fg,e^2=fi,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, 2661652);