Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7199982 |
| 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 | #31059 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1618 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16);(11,15)(13,16)(17,19,22,28)(18,23,25,32)(20,27,21,30)(24,31,29,26);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,5,6,4)(2,7,3,8)(11,15)(13,16)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7113299);