Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#121 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
475 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,10)(11,12)(13,14,17,18,16,19,15,20);(5,6,9,10)(7,11,8,12)(13,14,15,20,16,19,17,18);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,17)(14,18,19,20);(5,8,9,7)(6,11,10,12)(13,15,16,17)(14,18,19,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=e,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23086, #6382143, #6382146, … |
| maximal normal subgroup of | C2×#23086, #6382143, #6382146, … |
| maximal quotient of | #230806, #230807, #230808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16074);