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 | C42 | 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 ▷ C42 ▷ C22 ▷ C1 | |
2834 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)(3,9)(4,11)(7,14)(8,12)(13,15)(17,18,21,26,22,24,20,25)(19,27,28,23,30,32,29,31);(1,2,5,8)(3,7,9,15)(4,12,11,6)(10,14,16,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27);(1,3)(2,7)(4,10)(5,9)(6,13)(8,15)(11,16)(12,14)(18,24)(23,31)(25,26)(27,32);(3,9)(7,15)(10,16)(13,14);(17,20,22,21)(18,25,24,26)(19,28,30,29)(23,32,31,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=hi,f^2=i,g^2=hi,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, 6251272);