Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | D4 |
| commutator | C4 | C82 |
| Frattini, cosocle | C43 | C22 |
| socle, prime radical | C23 | #95 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,11,13,12,14,15,16);(2,5,7,6)(17,18,19,21,20,22,23,24)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,b^a=bc,c^a=cf,e^a=ef112 conjugacy classes — view the full character table.
| maximal subgroup of | #92862, #92863, C2×#7015, … |
| maximal normal subgroup of | #92862, #92863, C2×#7015, … |
| maximal quotient of | #92806, #92814, #93257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3);