Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | |
2146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,8,10)(7,11,12,9)(15,19)(18,20)(21,22)(23,28)(24,29)(25,30)(26,27)(31,36)(32,33)(34,35);(2,4)(13,14,17,16)(15,18,19,20)(22,27)(25,28)(29,35)(33,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,23,26,30)(22,28,27,25)(24,32,34,31)(29,33,35,36);(5,7,8,12)(6,9,10,11)(13,16,17,14)(15,20,19,18)(21,24,26,34)(22,29,27,35)(23,31,30,32)(25,33,28,36);(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,20)(17,19)(21,25,26,28)(22,30,27,23)(24,33,34,36)(29,32,35,31)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1035969);