Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470428 |
| 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 | #35328 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,27)(28,32)(30,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,27,31,22,28,21,24)(18,20,29,32,23,30,25,26);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,23)(19,28)(24,31)(25,29);(11,15)(13,16)(17,20,22,30)(18,24,23,31)(19,25,28,29)(21,26,27,32);(10,14)(13,16)(18,23)(20,30)(25,29)(26,32)Polycyclic presentation:
2^9|b^2=fg,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382736);