Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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×#2714 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2613 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)(7,11,8,10)(14,17)(15,19)(22,23)(24,31)(28,30)(32,33);(7,8)(10,11)(15,19)(18,20)(21,22,26,23)(24,29,31,27)(25,30,34,28)(32,35,33,36);(5,7,9,8)(6,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,31,29)(25,30,34,28)(32,35,33,36);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,15,16,19)(14,18,17,20)(21,24,26,31)(22,27,23,29)(25,33,34,32)(28,36,30,35);(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,25)(22,28)(23,30)(24,32)(26,34)(27,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 2978959);