Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25117 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1608 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16);(2,6)(7,8)(17,18,22,20)(19,26,28,32)(21,24,31,25)(23,30,29,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,22,28)(18,23,20,29)(21,30,31,27)(24,26,25,32);(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,26,28,32)(21,25,31,24)(23,30,29,27);(1,4,5,3)(2,8,6,7)(18,24)(20,25)(21,31)(23,26)(27,30)(29,32)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6494834);