Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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×C4⋊C4 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1608 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18,21,24,22,25,20,30)(19,26,29,31,27,32,28,23);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,21,29,22,27,20,28)(18,23,30,32,25,31,24,26);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,15)(13,16);(3,10)(8,15)(11,14)(12,16);(19,27)(23,31)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=g,b^2=fh,f^2=h,g^2=i,b^a=bf,f^a=fh,e^b=ei,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251633);