Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884809 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,13)(12,15)(14,16)(18,22)(20,27)(23,29)(25,28);(1,2)(3,8)(4,6)(5,7)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,30)(28,32);(1,3,5,4)(2,6,7,8)(10,13)(11,16)(14,15)(17,19,21,26)(18,23,22,29)(20,28,27,25)(24,32,30,31);(1,4,5,3)(2,6,7,8)(17,20,21,27)(18,24,22,30)(19,25,26,28)(23,31,29,32);(18,22)(23,29)(24,30)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6824485);