Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,9)(7,11)(15,19)(17,20)(21,22,26,24)(23,29,31,27)(25,28,34,32)(30,36,33,35);(2,4)(6,9)(10,12)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,36)(33,34);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,22)(23,27,31,29)(25,28,34,32)(30,36,33,35);(5,7)(6,10)(8,11)(9,12)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3009672);