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 | |
2128 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)(7,10)(8,12)(9,11)(15,19)(18,20)(21,22,26,27)(23,24,28,31)(25,29,35,36)(30,32,33,34);(1,2,3,4)(5,7,9,12)(6,10,11,8)(22,27)(24,31)(29,36)(32,34);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14,17,16)(15,18,19,20)(21,23,26,28)(22,24,27,31)(25,33,35,30)(29,34,36,32);(5,7)(6,8)(9,12)(10,11)(13,15,17,19)(14,18,16,20)(21,24)(22,28)(23,27)(25,34)(26,31)(29,30)(32,35)(33,36);(5,8)(6,7)(9,10)(11,12)(13,16,17,14)(15,20,19,18)(21,25,26,35)(22,29,27,36)(23,30,28,33)(24,32,31,34)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fgi,d^2=gi,e^2=fgi,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, 3702654);