Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,14,16,15)(17,18,22,27)(19,29,20,25)(21,24,28,23)(26,30,32,31);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,25)(20,22)(21,32)(23,31)(24,30)(26,28)(27,29);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,24,30,22,19,23,31)(18,26,21,29,27,32,28,25);(1,4,5,3)(2,6,8,7)(11,16)(14,15)(17,21,23,18,22,28,24,27)(19,29,30,26,20,25,31,32);(2,8)(6,7)(18,27)(21,28)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=fhi,d^2=fgi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510931);