Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(15,16)(18,19)(21,22)(23,26)(24,27)(25,30)(28,34)(29,31)(32,36)(33,35);(5,7,10,9)(6,11,8,12)(13,14,17,20)(15,18,16,19)(22,23)(24,31)(28,30)(32,33);(5,6,10,8)(7,12,9,11)(13,15,17,16)(14,18,20,19)(21,23,26,22)(24,29,31,27)(25,30,34,28)(32,36,33,35);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,31)(22,27,23,29)(25,33,34,32)(28,36,30,35);(5,9,10,7)(6,11,8,12)(13,16,17,15)(14,19,20,18)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=g,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2517476);