Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
2231 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)(7,11)(9,12)(15,19)(16,18)(21,22)(23,27)(24,26)(25,28)(29,34)(30,31)(32,33)(35,36);(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,16,19,18)(22,24)(27,31)(28,32)(34,35);(5,7)(6,9)(8,11)(10,12)(13,15,17,19)(14,18,20,16)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,22)(23,27,30,31)(25,28,33,32)(29,35,36,34);(5,7)(6,9)(8,11)(10,12)(13,16,17,18)(14,19,20,15)(21,25,26,33)(22,28,24,32)(23,29,30,36)(27,34,31,35)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2645646);