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 | |
2110 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)(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,20,16,19)(23,29)(24,31)(27,32)(28,33);(5,7,10,9)(6,11,12,8)(14,18)(19,20)(22,26)(27,32)(28,33)(34,36);(6,12)(8,11)(13,15,17,16)(14,19,18,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,16,17,15)(14,20,18,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36);(5,9,10,7)(6,8,12,11)(13,14)(15,19)(16,20)(17,18)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=f,d^2=f,e^2=fgh,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, 4853361);