Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(7,11)(9,12)(15,19)(17,20)(21,22)(23,26)(24,27)(25,30)(28,33)(29,31)(32,36)(34,35);(2,3)(13,14,16,18)(15,17,19,20)(22,23)(24,31)(25,33)(35,36);(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,18)(19,20)(21,23,26,22)(24,29,31,27)(25,30,33,28)(32,36,34,35);(2,3)(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,19)(18,20)(21,24)(22,27)(23,29)(25,34)(26,31)(28,36)(30,35)(32,33);(1,3,4,2)(21,25)(22,28)(23,30)(24,32)(26,33)(27,35)(29,36)(31,34)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fghi,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2859878);