Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) = C22×C4×#3003 (order 2048) · Inn(G) = C2×D4 · Out(G) = C25×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle, prime radical | C22 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,21,25,22,19,26,28,20,23,29,31,30,27,24,32);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(18,23)(19,27)(25,31)(28,32);(17,19,21,28,22,23,26,31,20,27,29,32,30,18,24,25)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=ef,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=ch,d^a=df,c^b=ch,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #93754, #93761, #122948, … |
| maximal normal subgroup of | #93754, #93761, #122948, … |
| maximal quotient of | #128678, #129140, #135895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5042);