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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,7)(8,12)(14,17)(15,19)(21,22,25,29)(23,27,31,26)(24,28,33,36)(30,35,32,34);(14,17)(18,20)(21,23,25,31)(22,26,29,27)(24,32,33,30)(28,35,36,34);(5,6,9,7)(8,10,12,11)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,33)(34,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,22)(23,26)(24,28)(25,29)(27,31)(30,34)(32,35)(33,36);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,33)(22,28,29,36)(23,30,31,32)(26,34,27,35)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=fgh,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, 994452);