Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | #28480 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1470 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)(11,15)(12,16)(13,14)(17,18,24,19)(20,23,26,22)(21,25,30,27)(28,29,32,31);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,23,26,24,18,22,20)(21,25,29,28,30,27,31,32);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,23,19,24,26,22,18)(21,28,29,27,30,32,31,25);(9,11,13,12)(10,15,14,16)(17,21,24,30)(18,25,19,27)(20,28,26,32)(22,31,23,29);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,26,19,20)(21,29,30,31)(25,28,27,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6393068);