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 | |
365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,24,21,19,20,23);(1,3)(2,5)(4,8)(6,7)(17,19,22,23,21,18,20,24);(3,8)(5,7)(10,13)(14,16)(18,19)(23,24);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,23,19,24)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22321, #6376761, #6376765, … |
| maximal normal subgroup of | C2×#22321, #6376761, #6376765, … |
| maximal quotient of | #355301, #355634, #355638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15309);