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 | #62800 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3078 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)(17,18)(19,26)(20,22)(21,25)(23,27)(24,29)(28,31)(30,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,22)(20,29)(23,30)(24,25)(26,31)(28,32);(2,8)(7,15)(9,10)(11,16)(17,19)(18,23)(20,28)(21,27)(22,30)(24,31)(25,26)(29,32);(19,27)(23,26)(28,32)(30,31);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)Polycyclic presentation:
2^9|a^2=g,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, 6250588);