Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
2177 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)(6,9)(10,12)(14,16)(15,19)(21,22,25,29)(23,27,31,26)(24,28,33,32)(30,35,36,34);(7,11)(10,12)(14,16)(18,20)(21,23,25,31)(22,26,29,27)(24,30,33,36)(28,34,32,35);(5,6,8,9)(7,10,11,12)(13,14,17,16)(15,20,19,18)(21,23)(22,27)(24,30)(25,31)(26,29)(28,35)(32,34)(33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20)(21,22)(23,26)(24,32)(25,29)(27,31)(28,33)(30,35)(34,36);(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,20,19,18)(21,24)(22,28)(23,30)(25,33)(26,34)(27,35)(29,32)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5537662);