Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | #25768 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 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)(11,15)(12,14)(13,16)(17,18,20,22)(19,25,27,23)(21,24,28,30)(26,31,29,32);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(18,22)(23,25)(24,30)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,19,20,27)(18,23,22,25)(21,29,28,26)(24,31,30,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(18,22)(23,25)(24,30)(31,32)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7496450);