Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,9)(10,12)(13,14,18,20)(15,16,19,17);(1,3,4,2)(5,6)(7,10)(8,9)(11,12);(5,7,8,11)(6,10,9,12)(13,15,17,14,18,19,16,20);(6,9)(10,12)(13,16,18,17)(14,19,20,15)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=ef,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23143, #6382595, #6382601, … |
| maximal normal subgroup of | C2×#23143, #6382595, #6382601, … |
| maximal quotient of | #231035, #231059, #231062, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16131);