Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2128 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)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,29)(24,26)(25,28)(27,30)(31,36)(32,33)(34,35);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20)(22,24)(23,30)(25,33)(35,36);(6,8)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,27)(24,29)(25,34)(26,30)(28,36)(31,33)(32,35);(1,2,3,4)(5,8,9,6)(7,11,12,10)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,36,34,35);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25,26,33)(22,28,24,32)(23,31,30,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5185964);