Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(6,7)(10,11)(13,14)(15,18)(16,19)(17,20)(21,22,26,27)(23,31,32,28)(24,34,35,29)(25,33,36,30);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(22,27)(23,32)(29,34)(30,33);(1,2,4,3)(21,23,26,32)(22,28,27,31)(24,33,35,30)(25,29,36,34);(1,3,4,2)(5,7,9,6)(8,10,12,11)(13,15)(14,18)(16,20)(17,19)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,28,36,31);(5,8)(6,10)(7,11)(9,12)(13,14)(15,17)(16,19)(18,20)(21,25,26,36)(22,30,27,33)(23,29,32,34)(24,31,35,28)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4347332);