Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| automorphism | Aut(G) order 4194304 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2130 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)(5,6,9,12)(7,8,11,10)(15,18)(16,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36);(5,7)(6,10)(8,12)(9,11)(13,14,17,19)(15,16,18,20)(22,26)(27,29)(28,32)(34,35);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,15,17,18)(14,16,19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8)(6,11)(7,12)(9,10)(13,16)(14,18)(15,19)(17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4217393);