Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1996 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)(5,6,9,8)(7,10,12,11)(13,14)(15,19)(16,17)(18,20)(21,22,26,29)(23,25,31,27)(24,32,35,28)(30,36,33,34);(6,8)(10,11)(13,15)(14,18)(16,19)(17,20)(21,23,26,31)(22,27,29,25)(24,33,35,30)(28,34,32,36);(5,7,9,12)(6,10,8,11)(13,15,17,20)(14,19,16,18)(21,23)(22,27)(24,33)(25,29)(26,31)(28,34)(30,35)(32,36);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,18)(17,20)(21,24,26,35)(22,28,29,32)(23,30,31,33)(25,34,27,36);(5,8,9,6)(7,11,12,10)(13,16,17,14)(15,18,20,19)(21,25,26,27)(22,23,29,31)(24,34,35,36)(28,30,32,33)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4406745);