Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(10,13)(11,15)(17,18,22,19,21,24,20,23);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,24,21,23,20,18);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,24,19);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(17,20,21,22)(18,19,24,23)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23282, #6383321, #6383326, … |
| maximal normal subgroup of | C2×#23282, #6383321, #6383326, … |
| maximal quotient of | #356445, #356788, #356832, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16270);