Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4452694 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(16,20)(18,19)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33);(2,3)(5,7)(6,9)(8,11)(10,12)(13,14,17,15)(16,18,20,19)(23,30)(26,29)(31,36)(33,35);(2,3)(6,8)(7,12)(13,15,17,14)(16,19,20,18)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36);(1,3,4,2)(5,8,10,6)(7,9,12,11)(13,16)(14,18)(15,19)(17,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,29)(32,36)(34,35);(5,9)(6,7)(8,12)(10,11)(21,24,25,32)(22,27,28,34)(23,31,30,36)(26,33,29,35)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=gi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4363415);