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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2172 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)(7,11,12,9)(13,14)(15,18)(16,20)(17,19)(21,22)(23,30)(24,25)(26,27)(28,33)(29,35)(31,32)(34,36);(13,15,17,20)(14,18,19,16)(22,27)(25,35)(28,30)(32,36);(5,7,8,12)(6,9,10,11)(13,15)(14,16)(17,20)(18,19)(21,23,26,33)(22,28,27,30)(24,34,29,31)(25,32,35,36);(1,2)(3,4)(21,24,26,29)(22,25,27,35)(23,31,33,34)(28,36,30,32);(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,29)(23,32)(24,27)(26,35)(28,31)(30,34)(33,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=gh,d^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, 993618);