Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,19,20,17)(22,26)(23,29)(24,32)(34,36);(2,4)(5,7)(6,10)(8,11)(9,12)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,2,3,4)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(5,7)(6,11)(8,10)(9,12)(13,15,16,20)(14,17,18,19)(21,22)(23,27)(24,33)(25,26)(28,32)(29,30)(31,36)(34,35);(5,8,9,6)(7,11,12,10)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=h,d^2=gi,e^2=fg,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, 2561883);