Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,21)(22,24)(25,26,28,31)(27,30,32,29);(10,12)(11,16)(17,18)(19,23)(20,21)(22,24)(25,27)(26,29)(28,32)(30,31);(10,12)(14,15)(18,21)(22,23)(25,26,28,31)(27,29,32,30);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266516, #72266532, #72266541, … |
| maximal normal subgroup of | #72266516, #72266532, #72266541, … |
| maximal quotient of | #72334835 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5093471);