Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #34909 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,20,23)(19,25,27,22)(21,24,28,31)(26,32,29,30);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(19,26)(21,28)(22,30)(24,31)(25,32)(27,29);(10,14)(13,15)(17,19,20,27)(18,22,23,25)(21,29,28,26)(24,32,31,30);(2,7)(3,8)(11,16)(13,15);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419015);