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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,7)(8,10,12,11)(13,14,18,20)(15,16,17,19)(21,22)(23,25)(24,29)(26,27)(28,30)(31,34)(32,36)(33,35);(1,2,3,4)(6,7)(10,11)(13,15)(14,19)(16,20)(17,18)(22,27)(23,30)(29,35)(31,36);(1,2)(3,4)(8,12)(10,11)(13,16,18,19)(14,17,20,15)(21,23)(22,28)(24,31)(25,27)(26,30)(29,32)(33,36)(34,35);(5,7,9,6)(8,10,12,11)(13,17,18,15)(14,16,20,19)(21,24)(22,29)(23,31)(25,34)(26,33)(27,35)(28,32)(30,36);(5,8)(6,10)(7,11)(9,12)(13,16)(14,17)(15,20)(18,19)(21,25,26,28)(22,30,27,23)(24,32,33,34)(29,31,35,36)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=i,d^2=fg,e^2=g,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, 3341138);