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 | #62805 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2890 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(18,22)(19,25)(20,27)(23,30)(24,28)(29,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,18,21,24)(19,23,25,31)(20,28,27,22)(26,30,32,29);(2,8)(7,15)(9,10)(11,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,31)(27,32)(28,30);(19,25)(23,31)(26,32)(29,30);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=hi,f^2=i,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,d^c=dh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250589);