Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2099 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)(8,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,29)(25,28)(27,31)(30,33)(32,35)(34,36);(5,7,10,8)(6,11,12,9)(13,15,17,20)(14,18,16,19)(22,23)(24,31)(25,33)(34,35);(6,12)(7,8)(14,16)(18,19)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35);(1,2,3,4)(5,8,10,7)(6,11,12,9)(13,16,17,14)(15,19,20,18)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(5,9,10,11)(6,7,12,8)(13,15)(14,18)(16,19)(17,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)Polycyclic presentation:
2^9|b^2=fg,c^2=f,d^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188521);