Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,10)(3,5,9,6)(4,8,12,15)(11,13,16,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,6,2,7,9,5,10)(4,11,14,8,12,16,13,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4)(2,8)(3,11)(5,13)(6,14)(7,12)(9,16)(10,15)(19,25)(22,28)(26,31)(29,32);(4,12)(8,15)(11,16)(13,14);(1,5,7,6)(2,9,10,3)(4,13,12,14)(8,16,15,11)Polycyclic presentation:
2^9|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,c^b=ci,e^b=eh,d^c=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251536);