Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1920 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)(6,7)(8,12)(13,14,16,18)(15,19,20,17)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,36)(34,35);(5,6,9,7)(8,10,12,11)(13,14,16,18)(15,17,20,19)(22,27)(23,28)(29,35)(30,33);(8,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,28)(22,24,27,32)(25,33,34,30)(29,36,35,31);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,17)(16,18)(19,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,33)(30,35)(34,36);(5,8,9,12)(6,10,7,11)(13,15)(14,17)(16,20)(18,19)(21,25,26,34)(22,29,27,35)(23,30,28,33)(24,31,32,36)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=gh,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4509111);