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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2467 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)(7,11)(13,14,18,17)(15,19,20,16)(23,29)(27,31)(30,35)(34,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22,26,24)(23,27,29,31)(25,32,33,28)(30,36,35,34);(7,11)(10,12)(13,16,18,19)(14,15,17,20)(21,23)(22,27)(24,31)(25,30)(26,29)(28,34)(32,36)(33,35);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,19,20,16)(21,24,26,22)(23,27,29,31)(25,32,33,28)(30,34,35,36);(5,6)(7,12)(8,9)(10,11)(13,15)(14,19)(16,17)(18,20)(21,25,26,33)(22,28,24,32)(23,30,29,35)(27,34,31,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=fg,e^2=g,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, 5518891);