Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(18,22)(20,28)(23,29)(24,30)(25,32)(27,31);(9,10)(11,14)(12,13)(15,16)(17,18,20,22,21,23,28,29)(19,25,27,32,26,24,31,30);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,23)(19,26)(22,29)(27,31);(10,13)(14,16)(17,19)(18,24)(20,27)(21,26)(22,30)(23,25)(28,31)(29,32);(1,4,5,3)(2,6,7,8)(9,11,12,15)(10,14,13,16)(19,26)(24,25)(27,31)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,e^2=ghi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6376787);