Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993915 |
| 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 | #39148 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(11,15)(14,16)(17,18,20,24,21,22,29,31)(19,25,28,30,27,32,26,23);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(18,22)(23,30)(24,31)(25,32);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,23,22,30)(20,26,29,28)(24,32,31,25);(3,8)(6,7)(10,13)(14,16)(17,20,21,29)(18,24,22,31)(19,26,27,28)(23,32,30,25);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=fgi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7057252);