Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,8,5)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,22)(21,26)(23,30)(24,27)(28,32)(29,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,28)(18,25,27,32)(20,29,30,26)(21,22,31,23);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,20,22,18,24,30,23,27)(19,26,31,32,28,29,21,25);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(17,21)(18,26)(19,22)(20,25)(23,28)(24,31)(27,29)(30,32);(1,5,6,4)(2,3,7,8)(17,22,24,23)(18,20,27,30)(19,21,28,31)(25,26,32,29)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897277);