Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1953 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)(15,19)(18,20)(22,24)(25,33)(27,31)(29,35);(8,12)(10,11)(14,17)(18,20)(21,22,26,24)(23,27,30,31)(25,28,33,32)(29,34,35,36);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,36)(33,35);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(21,24,26,22)(23,27,30,31)(25,28,33,32)(29,36,35,34);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,28)(23,29)(24,32)(26,33)(27,34)(30,35)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=h,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5517325);