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 | |
2002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,3)(5,6,10,8)(7,12,11,9)(14,17)(15,19)(22,26)(24,32)(27,29)(30,36);(14,17)(18,20)(21,22,25,26)(23,27,31,29)(24,33,32,28)(30,35,36,34);(1,2,4,3)(5,7)(6,9)(8,12)(10,11)(13,14,16,17)(15,18,19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,2,4,3)(5,8,10,6)(7,12,11,9)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(1,3,4,2)(5,9,10,12)(6,11,8,7)(13,15,16,19)(14,18,17,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=fhi,d^2=fgh,e^2=g,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, 1085611);