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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(7,11)(10,12)(14,17)(15,19)(21,22,26,27)(23,28,29,24)(25,30,35,36)(31,34,33,32);(1,2,3,4)(15,19)(18,20)(22,27)(24,28)(30,36)(32,34);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,29)(22,28,27,24)(25,33,35,31)(30,32,36,34);(5,7,9,11)(6,10,8,12)(13,15,16,19)(14,18,17,20)(21,24,26,28)(22,29,27,23)(25,34,35,32)(30,31,36,33);(5,8,9,6)(7,12,11,10)(13,14)(15,18)(16,17)(19,20)(21,25,26,35)(22,30,27,36)(23,31,29,33)(24,32,28,34)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gi,d^2=i,e^2=gi,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, 3702378);