Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2315 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)(3,6)(4,5)(7,8)(17,18,22,24)(19,25,27,26)(20,21,29,30)(23,31,32,28);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,28,22,27,30,31)(18,23,20,25,24,32,29,26);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,21,24,22,29,30,18)(19,26,28,23,27,25,31,32);(11,16)(13,15);(10,14)(13,15)(19,27)(23,32)(25,26)(28,31)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413632);