Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10,11,13,12,14,15,16)(17,18)(19,26)(20,22)(21,23)(24,28)(25,29)(27,31)(30,32);(1,3,5,4)(2,6,7,8)(18,22)(19,27)(20,29)(23,25)(24,26)(28,30);(1,4,5,3)(2,6,7,8)(18,23)(22,25)(24,26)(31,32);(17,19,21,28)(18,24,23,26)(20,30,29,27)(22,31,25,32)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,e^d=eh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 133697);