Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| 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×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1474 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)(4,6)(5,7)(17,18,22,24,21,25,20,26)(19,23,29,30,27,31,28,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,27)(18,23,25,31)(20,28,22,29)(24,30,26,32);(9,11)(10,13)(12,16)(14,15)(18,24)(20,22)(23,30)(25,26)(28,29)(31,32);(11,16)(13,15)(18,25)(19,27)(24,26)(28,29);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,25)(23,31)(24,26)(30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413575);