Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232584 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,7)(5,10)(6,9)(8,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(2,7)(3,12)(4,5)(9,10)(11,16)(13,14)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30);(2,9)(7,10)(8,15)(11,16)Polycyclic presentation:
2^9|b^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139786);