Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(6,7)(8,12)(15,19)(17,20)(21,22,25,26)(23,27,29,34)(24,31,33,28)(30,36,32,35);(6,7)(10,11)(13,14,16,18)(15,17,19,20)(22,26)(23,29)(28,31)(30,32);(5,6,9,7)(8,10,12,11)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,28)(25,26)(29,34)(30,35)(31,33)(32,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,23,25,29)(22,27,26,34)(24,32,33,30)(28,36,31,35);(5,8)(6,10)(7,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,24,25,33)(22,28,26,31)(23,30,29,32)(27,35,34,36)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4774591);