Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2386 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)(5,6,9,12)(7,11,8,10)(14,16)(15,20)(21,22)(23,27)(24,26)(25,28)(29,30)(31,35)(32,34)(33,36);(5,6)(7,10)(8,11)(9,12)(13,14,17,16)(15,19,20,18)(22,24)(23,29)(28,32)(31,33);(5,7,9,8)(6,10,12,11)(13,15,17,20)(14,18,16,19)(21,23,26,29)(22,27,24,30)(25,33,34,31)(28,36,32,35);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,16,17,14)(15,19,20,18)(21,24,26,22)(23,30,29,27)(25,28,34,32)(31,35,33,36);(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,25,26,34)(22,28,24,32)(23,31,29,33)(27,35,30,36)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3243592);