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 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,21)(20,28)(22,24)(23,26)(25,30)(27,29)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,26)(25,28)(27,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,18,22,26,23,24)(20,27,31,25,30,32,29,28);(2,6)(5,7)(11,16)(13,15)(17,20)(18,25)(19,27)(21,31)(22,30)(23,29)(24,28)(26,32);(10,14)(13,15)(17,21,22,23)(18,19,24,26)(20,29,30,31)(25,32,28,27)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6815434);