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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(7,11)(10,12)(13,14,16,18)(15,17,19,20)(21,22)(23,31)(24,33)(25,35)(26,27)(28,32)(29,34)(30,36);(5,6,9,8)(7,10,11,12)(15,19)(17,20)(22,27)(23,32)(29,33)(30,35);(1,2,4,3)(13,15)(14,17)(16,19)(18,20)(21,23)(22,28)(24,30)(25,33)(26,32)(27,31)(29,36)(34,35);(1,3,4,2)(5,7)(6,10)(8,12)(9,11)(21,24,26,34)(22,29,27,33)(23,30,32,35)(25,28,36,31);(5,8,9,6)(7,12,11,10)(13,14)(15,17)(16,18)(19,20)(21,25,26,36)(22,30,27,35)(23,33,32,29)(24,31,34,28)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fg,d^2=g,e^2=fgi,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, 4355935);