Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,10)(7,12)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(24,32)(34,36);(15,19)(17,20)(21,22,25,26)(23,27,29,30)(24,33,32,28)(31,36,35,34);(5,6)(7,11)(8,12)(9,10)(13,15)(14,17)(16,19)(18,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(1,2,3,4)(5,7,9,12)(6,11,10,8)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,8,9,11)(6,12,10,7)(13,14)(15,17)(16,18)(19,20)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=f,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4248689);