Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
2141 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,18)(21,22,26,29)(23,24,31,27)(25,28,34,36)(30,32,33,35);(6,9)(11,12)(14,18)(16,19)(21,23,26,31)(22,27,29,24)(25,33,34,30)(28,32,36,35);(5,7,10,8)(6,11,9,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,36)(33,34);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,15)(14,16)(17,20)(18,19)(21,24,26,27)(22,23,29,31)(25,32,34,35)(28,30,36,33);(5,9,10,6)(7,12,8,11)(13,16)(14,15)(17,19)(18,20)(21,25,26,34)(22,28,29,36)(23,30,31,33)(24,32,27,35)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1947093);