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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2013 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)(7,11)(13,14,17,18)(15,19,20,16)(21,22)(23,24)(25,33)(26,27)(28,30)(29,36)(31,35)(32,34);(5,6,9,10)(7,12,11,8)(14,18)(16,19)(22,27)(23,30)(29,33)(31,34);(6,10)(8,12)(13,15,17,20)(14,19,18,16)(21,23)(22,28)(24,27)(25,34)(26,30)(29,32)(31,36)(33,35);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,16)(14,15)(17,19)(18,20)(21,24,26,28)(22,23,27,30)(25,35,36,32)(29,34,33,31);(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,18)(17,20)(21,25,26,36)(22,29,27,33)(23,31,30,34)(24,32,28,35)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fh,d^2=h,e^2=g,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, 3255854);