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) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,10)(7,8)(15,19)(18,20)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,35,36,33);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(14,17)(18,20)(22,27)(23,30)(29,32)(31,36);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,31)(25,27)(26,30)(29,33)(32,35)(34,36);(5,8,9,7)(6,11,10,12)(13,15)(14,18)(16,19)(17,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,6)(7,11)(8,12)(9,10)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,36,32,31)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=g,d^2=f,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, 2524440);