Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1562 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,22,31,21,27,20,26)(19,25,28,23,30,24,29,32);(2,6)(7,8)(9,11,14,10)(12,15,13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,26)(25,31)(27,32);(1,3,5,4)(2,7,6,8)(10,11)(15,16)(17,19)(18,24)(20,29)(21,30)(22,28)(23,26)(25,27)(31,32);(9,12,14,13)(10,15,11,16)(17,19)(18,25)(20,29)(21,30)(22,28)(23,31)(24,27)(26,32);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(17,20,21,22)(18,26,27,31)(19,28,30,29)(23,24,32,25)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398025);