Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,7)(8,12)(13,14,16,18)(15,17,19,20)(21,22,27,30)(23,31,26,28)(24,29,34,25)(32,35,36,33);(6,7)(10,11)(15,19)(17,20)(21,23,27,26)(22,28,30,31)(24,32,34,36)(25,35,29,33);(5,6,9,7)(8,10,12,11)(13,15,16,19)(14,17,18,20)(21,24)(22,29)(23,32)(25,30)(26,36)(27,34)(28,33)(31,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,25,27,29)(22,24,30,34)(23,33,26,35)(28,36,31,32);(5,8,9,12)(6,10,7,11)(21,26,27,23)(22,28,30,31)(24,32,34,36)(25,33,29,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=h,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3245860);