Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
2321 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)(6,7)(10,11)(14,18)(16,19)(21,22,25,26)(23,29,30,27)(24,28,32,34)(31,36,33,35);(5,6)(7,9)(8,10)(11,12)(13,14,17,18)(15,16,20,19)(22,26)(23,30)(24,32)(35,36);(6,7)(8,12)(15,20)(16,19)(21,23,25,30)(22,27,26,29)(24,33,32,31)(28,36,34,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,17,20)(14,19,18,16)(21,22)(23,27)(24,28)(25,26)(29,30)(31,35)(32,34)(33,36);(5,8,9,12)(6,10,7,11)(13,16,17,19)(14,20,18,15)(21,24)(22,28)(23,31)(25,32)(26,34)(27,35)(29,36)(30,33)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=g,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5120127);