Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994171 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,24,20,25,21,26)(19,27,30,32,28,23,29,31);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,20,28)(18,23,25,27)(21,29,22,30)(24,31,26,32);(9,11)(10,13)(12,16)(14,15)(18,24)(21,22)(23,31)(25,26)(27,32)(29,30);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(19,28)(23,27)(29,30)(31,32);(10,14)(13,15)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914965);