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) = C26 |
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×#546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,18)(20,24)(21,23);(1,2)(3,4)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,19,24,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,24)(19,22);(2,4)(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,23,24,19)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22424, #6379051, #6379060, … |
| maximal normal subgroup of | C2×#22424, #6379051, #6379060, … |
| maximal quotient of | #220061, #220064, #220065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15412);