Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,20)(12,16)(13,18)(14,19)(15,23)(17,22)(21,24);(1,2,3,4)(9,11,14,12)(10,16,19,20)(13,22,15,21)(17,18,24,23);(9,12,14,11)(10,17,19,24)(13,22,15,21)(16,18,20,23);(5,7,8,6)(9,13,14,15)(10,18,19,23)(11,21,12,22)(16,17,20,24)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,d^2=gh,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21512, #6372803, #6372810, … |
| maximal normal subgroup of | C2×#21512, #6372803, #6372810, … |
| maximal quotient of | #210455, #210463, #210495, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14500);