Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,31,22,26,21,27)(19,25,30,24,20,29,28,32);(2,5)(6,7)(10,14)(11,12)(17,19)(18,24)(20,22)(21,30)(23,28)(25,31)(26,32)(27,29);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,19)(18,25,26,29)(21,30,23,28)(24,27,32,31);(9,12,13,11)(10,15,14,16)(19,20)(24,32)(25,29)(28,30);(18,26)(24,32)(25,29)(27,31)Polycyclic presentation:
2^9|a^2=fg,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370857);