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) = C26×C4 |
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 | |
1818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6)(7,9)(8,12)(10,11)(15,19)(18,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(6,11)(7,8)(14,17)(18,20)(21,22,25,28)(23,26,30,34)(24,27,32,36)(29,33,31,35);(5,7,10,8)(6,12,11,9)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,35)(32,36);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,15,16,19)(14,18,17,20)(21,23,25,30)(22,26,28,34)(24,31,32,29)(27,35,36,33);(5,9,10,12)(6,8,11,7)(21,24,25,32)(22,27,28,36)(23,29,30,31)(26,33,34,35)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=g,d^2=h,e^2=f,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, 2517212);