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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,10)(7,11,12,9)(14,16)(15,19)(22,26)(23,29)(28,32)(31,33);(7,12)(9,11)(15,19)(18,20)(21,22,25,26)(23,30,29,27)(24,32,34,28)(31,35,33,36);(5,7)(6,9)(8,12)(10,11)(13,14,17,16)(15,18,19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,34)(33,35);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,19)(14,18,16,20)(21,23,25,29)(22,27,26,30)(24,33,34,31)(28,36,32,35);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,31)(25,34)(26,32)(27,35)(29,33)(30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=gh,d^2=fgh,e^2=gh,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, 2992756);