Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,16)(11,13,12,15)(17,18,21,24)(19,20,23,22)(27,31)(29,32);(2,3)(4,8)(9,11)(10,15)(12,14)(13,16)(17,19,21,23)(18,22,24,20)(25,26,28,30)(27,29,31,32);(2,3)(6,7)(10,16)(11,12)(17,20,21,22)(18,23,24,19);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,16,15)(17,20)(18,19)(21,22)(23,24)(25,27,28,31)(26,29,30,32);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=fg,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419641);