Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
2192 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)(5,6,9,11)(7,8,12,10)(13,14,17,18)(15,19,20,16)(21,22)(23,30)(24,25)(26,28)(27,29)(31,35)(32,33)(34,36);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,23,27,24)(22,25,29,30)(26,31,34,32)(28,33,36,35);(6,11)(8,10)(14,18)(16,19)(21,24,27,23)(22,25,29,30)(26,31,34,32)(28,35,36,33);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,20)(14,19,18,16)(21,25)(22,23)(24,29)(26,33)(27,30)(28,31)(32,36)(34,35);(5,8,9,10)(6,7,11,12)(13,16,17,19)(14,20,18,15)(21,26,27,34)(22,28,29,36)(23,31,24,32)(25,33,30,35)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2023904);