Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2168 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)(5,6,9,10)(7,12,11,8)(15,19)(18,20)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(5,6,9,10)(7,8,11,12)(14,17)(18,20)(22,25)(27,29)(28,32)(34,35);(2,3)(6,10)(7,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,35)(33,36);(1,3,4,2)(5,7,9,11)(6,8,10,12)(21,24,26,33)(22,28,25,32)(23,30,31,36)(27,34,29,35);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25,26,22)(23,29,31,27)(24,32,33,28)(30,35,36,34)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4855781);