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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,10)(8,12)(9,11)(15,19)(18,20)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,36,34,35);(2,4)(5,7,9,12)(6,10,11,8)(14,17)(18,20)(22,23)(25,33)(27,29)(31,34);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14,16,17)(15,18,19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,35,34,36);(5,7)(6,8)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,24,26,32)(22,27,23,29)(25,34,33,31)(28,36,30,35);(5,8)(6,7)(9,10)(11,12)(21,25,26,33)(22,28,23,30)(24,31,32,34)(27,35,29,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgi,d^2=f,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, 3009734);