Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
2014 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)(7,10)(8,12)(9,11)(21,22)(23,31)(24,25)(26,27)(28,34)(29,30)(32,35)(33,36);(5,7,9,12)(6,10,11,8)(13,14,16,18)(15,17,19,20)(22,27)(24,30)(28,31)(32,36);(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,34)(22,28,27,31)(24,32,30,36)(25,33,29,35);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(21,24)(22,29)(23,32)(25,27)(26,30)(28,35)(31,33)(34,36);(5,8)(6,7)(9,10)(11,12)(13,15,16,19)(14,17,18,20)(21,25,26,29)(22,30,27,24)(23,33,34,35)(28,36,31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=f,e^2=gh,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, 2521389);