Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,10)(7,11)(14,15)(16,20)(21,22,27,32)(23,30,25,28)(24,29,31,26)(33,34,35,36);(5,6,9,10)(7,8,11,12)(14,15)(18,19)(21,23,27,25)(22,28,32,30)(24,33,31,35)(26,34,29,36);(6,10)(8,12)(13,14,17,15)(16,18,20,19)(21,24)(22,29)(23,33)(25,35)(26,32)(27,31)(28,36)(30,34);(1,2,3,4)(5,7,9,11)(6,8,10,12)(13,15,17,14)(16,19,20,18)(21,25,27,23)(22,30,32,28)(24,35,31,33)(26,36,29,34);(5,8)(6,11)(7,10)(9,12)(13,16,17,20)(14,18,15,19)(21,26,27,29)(22,31,32,24)(23,34,25,36)(28,35,30,33)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4576975);