Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62748 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,11)(5,10)(6,9)(7,12)(13,16)(14,15)(17,18,22,28,23,26,21,27)(19,29,30,20,25,32,31,24);(1,3)(2,7)(4,12)(5,14)(6,13)(8,10)(9,15)(11,16)(17,19)(18,24)(20,26)(21,31)(22,30)(23,25)(27,32)(28,29);(1,3)(2,8)(4,14)(5,12)(6,13)(7,10)(9,16)(11,15)(17,20)(18,25)(19,26)(21,29)(22,32)(23,24)(27,30)(28,31);(2,9)(7,15)(8,16)(10,11)(18,26)(20,24)(27,28)(29,32);(1,4,6,5)(2,10,9,11)(3,12,13,14)(7,8,15,16)(17,21,23,22)(18,27,26,28)(19,30,25,31)(20,32,24,29)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250475);