Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,17,14,11,18,16)(12,15,23,24,21,19,22,20);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,20)(13,16)(15,23)(17,18)(19,22)(21,24);(2,6)(5,8)(12,21)(15,19)(20,24)(22,23);(9,12)(10,15)(11,19)(13,23)(14,21)(16,20)(17,24)(18,22)Polycyclic presentation:
2^8|a^2=g,b^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,d^c=dh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22588, #6379743, #6379744, … |
| maximal normal subgroup of | C2×#22588, #6379743, #6379744, … |
| maximal quotient of | #224954, #224955, #224956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15576);