Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(8,12)(13,14,17,15)(16,19,20,18)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36);(2,4)(5,6,9,10)(7,12,11,8)(22,26)(27,29)(28,32)(35,36);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,15,17,14)(16,19,20,18)(21,22)(23,27)(24,32)(25,26)(28,34)(29,31)(30,36)(33,35);(5,6)(7,8)(9,10)(11,12)(13,16,17,20)(14,18,15,19)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35);(5,8)(6,7)(9,12)(10,11)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fghi,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3455783);