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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,10)(7,12)(13,14,16,18)(15,19,20,17)(21,22)(23,26)(24,32)(25,28)(27,34)(29,31)(30,36)(33,35);(2,4)(5,6)(7,8)(9,10)(11,12)(21,22,25,28)(23,26,31,29)(24,32,34,27)(30,36,33,35);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(21,22)(23,29)(24,27)(25,28)(26,31)(30,36)(32,34)(33,35);(5,7)(6,11)(8,10)(9,12)(13,15,16,20)(14,17,18,19)(21,23,25,31)(22,26,28,29)(24,33,34,30)(27,36,32,35);(5,8,9,11)(6,12,10,7)(21,24,25,34)(22,27,28,32)(23,30,31,33)(26,35,29,36)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=g,d^2=fi,e^2=f,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, 3050743);