Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
2054 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)(5,6,9,10)(7,12,11,8)(15,19)(18,20)(21,22)(23,27)(24,32)(25,26)(28,31)(29,34)(30,36)(33,35);(6,10)(7,11)(14,17)(18,20)(21,23,26,31)(22,27,25,28)(24,30,34,33)(29,35,32,36);(7,11)(8,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,28)(24,33)(25,27)(26,31)(29,35)(30,34)(32,36);(1,2,3,4)(5,7,9,11)(6,8,10,12)(21,24)(22,29)(23,30)(25,32)(26,34)(27,35)(28,36)(31,33);(5,8)(6,11)(7,10)(9,12)(13,15,16,19)(14,18,17,20)(21,25,26,22)(23,27,31,28)(24,32,34,29)(30,35,33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188764);