Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
2032 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)(6,8)(7,11)(14,18)(16,19)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(7,11)(10,12)(14,18)(15,20)(21,22,25,28)(23,26,30,34)(24,27,32,36)(29,33,31,35);(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15,17,20)(14,19,18,16)(21,23,25,30)(22,26,28,34)(24,31,32,29)(27,35,36,33);(5,8,9,6)(7,12,11,10)(13,16,17,19)(14,20,18,15)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,33,34,35)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=fgh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2518992);