Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
2272 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)(5,6,8,10)(7,9,11,12)(13,14,17,18)(15,16,19,20)(21,22)(23,27)(24,26)(25,28)(29,35)(30,31)(32,34)(33,36);(5,7,8,11)(6,9,10,12)(13,15,17,19)(14,16,18,20)(22,24)(27,31)(28,32)(35,36);(6,10)(9,12)(14,18)(15,19)(21,23,26,30)(22,27,24,31)(25,33,34,29)(28,36,32,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36);(5,7)(6,9)(8,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4943297);