Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(4,8)(10,14)(12,15)(17,18,21,22)(19,23,24,20)(25,26,29,31)(27,28,32,30);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,24)(18,20,22,23)(25,27,29,32)(26,30,31,28);(4,8)(6,7)(10,14)(12,15)(18,22)(19,24)(25,28,29,30)(26,27,31,32);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,20)(18,19)(21,23)(22,24)(25,26)(27,28)(29,31)(30,32);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,16,14,11)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fhi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 948027);