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 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,7)(8,10,12,11)(13,14,19,20)(15,18,17,16)(23,29)(24,31)(27,32)(28,33);(13,15,19,17)(14,16,20,18)(22,26)(27,32)(28,33)(34,36);(6,7)(10,11)(13,16,19,18)(14,15,20,17)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,17,19,15)(14,16,20,18)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,8,9,12)(6,10,7,11)(13,18,19,16)(14,15,20,17)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fh,d^2=fgh,e^2=fg,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, 4843077);