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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,11)(8,10)(9,12)(13,14,17,16)(15,19,20,18)(21,22)(23,25)(24,32)(26,27)(28,30)(29,34)(31,33)(35,36);(2,4)(5,7,10,9)(6,11,8,12)(22,27)(23,30)(29,32)(31,36);(1,2)(3,4)(5,6,10,8)(7,12,9,11)(13,14,17,16)(15,18,20,19)(21,23)(22,28)(24,31)(25,27)(26,30)(29,33)(32,35)(34,36);(5,8,10,6)(7,11,9,12)(13,15,17,20)(14,18,16,19)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,9,10,7)(6,11,8,12)(13,16,17,14)(15,19,20,18)(21,25,26,28)(22,30,27,23)(24,33,34,35)(29,31,32,36)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fi,d^2=i,e^2=i,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, 3732559);