Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,16,12,15)(18,22)(19,26)(20,23)(24,31)(25,29)(28,32);(2,5)(6,7)(10,14)(11,12)(17,18,23,29,21,25,20,22)(19,24,30,31,26,32,27,28);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,19,21,26)(18,24,25,32)(20,27,23,30)(22,28,29,31);(9,12,13,11)(10,15,14,16);(19,26)(24,32)(27,30)(28,31)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371339);