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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,21)(18,24)(22,27)(25,28);(5,8,9,7)(6,10,11,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(2,4)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,g^2=h,b^a=be,c^a=cf,c^b=ch,d^b=de,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22756, #6380450, #6380453, … |
| maximal normal subgroup of | C2×#22756, #6380450, #6380453, … |
| maximal quotient of | #226710, #226711, #226712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15744);