Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1885 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)(7,10)(8,12)(9,11)(13,14)(15,20)(16,18)(17,19)(21,22,26,29)(23,27,24,31)(25,28,33,36)(30,34,32,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,20)(17,18)(21,23,26,24)(22,27,29,31)(25,32,33,30)(28,35,36,34);(6,11)(7,12)(13,14,18,16)(15,19,17,20)(21,24,26,23)(22,27,29,31)(25,32,33,30)(28,34,36,35);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,16,18,14)(15,19,17,20)(21,22)(23,27)(24,31)(25,28)(26,29)(30,34)(32,35)(33,36);(5,8)(6,7)(9,10)(11,12)(13,17,18,15)(14,19,16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,36)(31,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=gh,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989878);