Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1393 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)(18,23)(19,27)(21,24)(25,32)(26,30)(28,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,24,30,22,26,21,23)(19,25,31,20,28,32,27,29);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19,21,27,22,28,24,31)(18,25,23,29,26,32,30,20);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,22,29)(18,19,26,28)(21,25,24,32)(23,27,30,31);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(18,26)(20,29)(23,30)(25,32)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6448171);