Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,9)(11,12)(14,17)(15,19)(21,22)(23,29)(24,32)(25,30)(26,27)(28,34)(31,35)(33,36);(5,6,10,8)(7,11,9,12)(14,17)(18,20)(22,27)(23,30)(28,32)(31,36);(5,7,10,9)(6,11,8,12)(13,14,16,17)(15,18,19,20)(21,23)(22,25)(24,31)(26,30)(27,29)(28,35)(32,33)(34,36);(1,2,3,4)(5,8,10,6)(7,12,9,11)(21,24,26,34)(22,28,27,32)(23,31,30,36)(25,35,29,33);(5,9,10,7)(6,12,8,11)(13,15,16,19)(14,18,17,20)(21,25)(22,23)(24,33)(26,29)(27,30)(28,36)(31,32)(34,35)Polycyclic presentation:
2^9|b^2=gh,c^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1203225);