Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62815 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(18,23,25,29)(24,30,31,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,30)(28,32);(2,8)(7,15)(9,13)(10,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,30)(25,31)(29,32);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(19,26)(24,31)(27,28)(30,32);(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,30,31,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=ch,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250680);