Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,7)(9,10,14,12)(11,15,13,16)(19,23)(20,24)(27,31)(29,32);(1,2,5,6)(3,8,7,4)(9,11,14,13)(10,15,12,16)(17,18,21,22)(19,24,23,20);(2,6)(4,8)(9,11,14,13)(10,16,12,15)(17,19)(18,20)(21,23)(22,24)(25,26,28,30)(27,29,31,32);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,15,13,16)(17,18)(19,20)(21,22)(23,24)(25,27,28,31)(26,29,30,32);(1,4)(2,3)(5,8)(6,7)(9,13,14,11)(10,15,12,16)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=ghi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3453540);