Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| 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 | C42 | 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 ▷ C42 ▷ C22 ▷ C1 | |
3010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,22,27,23,28,21,26)(19,29,30,24,20,25,31,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,20)(18,24,28,32)(21,30,22,31)(25,27,29,26);(17,20,23,19)(18,25,28,29)(21,30,22,31)(24,27,32,26);(17,21,23,22)(18,26,28,27)(19,30,20,31)(24,25,32,29);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bf,f^a=fh,f^b=fi,g^b=ghi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251349);