Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| 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 | C22×C4×D4 |
| prime radical | #60954 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1602 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)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18,22,25,23,19,21,26)(20,24,28,32,29,27,30,31);(1,3)(2,7)(4,12)(5,14)(6,13)(8,11)(9,15)(10,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,29)(28,31)(30,32);(17,20,22,28,23,29,21,30)(18,24,25,32,19,27,26,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(17,21,23,22)(18,25,19,26)(20,28,29,30)(24,31,27,32);(2,9)(7,15)(8,10)(11,16)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,d^b=di,f^b=fh,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252691);