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 | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
522 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)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22);(1,2,3,4)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18);(5,7,8,6)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=h,d^2=g,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×#21515, #6372803, #6372810, … |
| maximal normal subgroup of | C2×#21515, #6372803, #6372810, … |
| maximal quotient of | #210458, #210466, #210498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14503);