Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,29,32,25)(20,22,28,24)(21,26,31,30);(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,25,27,29)(20,26,28,30)(21,24,31,22);(9,11)(10,13)(12,16)(14,15)(17,20,24,18,23,28,22,27)(19,30,31,29,32,26,21,25);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,21,23,31)(18,26,27,30)(19,22,32,24)(20,29,28,25);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,22,23,24)(18,20,27,28)(19,31,32,21)(25,30,29,26)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6936323);