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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2256 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)(5,6,9,11)(7,8,12,10)(14,18)(15,19)(21,22,27,31)(23,29,24,28)(25,36,30,26)(32,35,34,33);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,18)(19,20)(21,23,27,24)(22,28,31,29)(25,34,30,32)(26,33,36,35);(6,11)(7,12)(14,18)(16,20)(21,24,27,23)(22,29,31,28)(25,34,30,32)(26,33,36,35);(1,2,3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,25,27,30)(22,26,31,36)(23,32,24,34)(28,35,29,33);(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,26)(22,30)(23,33)(24,35)(25,31)(27,36)(28,32)(29,34)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=hi,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, 4575298);