Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62752 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10,7,4)(5,8)(6,9)(11,16,14,12)(13,15)(18,23,25,28)(24,31,32,30);(1,3,6,7)(2,4,9,10)(5,11,13,14)(8,12,15,16)(17,18)(19,24)(20,23)(21,28)(22,25)(26,30)(27,32)(29,31);(1,4)(2,7)(3,9)(5,12)(6,10)(8,14)(11,15)(13,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31);(1,5,6,13)(2,8,9,15)(3,11,7,14)(4,12,10,16)(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94354);