Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6636868 |
| 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 | #29742 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1581 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,31,30,27);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,22)(19,26)(24,29)(27,31);(11,15)(13,16)(17,19,21,27,20,26,28,31)(18,23,24,25,22,30,29,32);(2,6)(3,8)(11,15)(13,16)(18,22)(23,30)(24,29)(25,32);(2,6)(5,7)(10,14)(13,16)(19,26)(23,30)(25,32)(27,31)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6544303);