Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(16,23)(19,24)(20,25)(22,27)(26,28);(2,4)(13,14,16,18,17,20,23,25)(15,19,22,24,21,26,27,28);(6,9)(10,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(5,7,8,11)(6,10,9,12)(15,21)(19,26)(22,27)(24,28)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=f,e^2=h,b^a=be,c^a=cf,e^a=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21412, #6372438, #6372444, … |
| maximal normal subgroup of | C2×#21412, #6372438, #6372444, … |
| maximal quotient of | #207552, #207567, #207582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14400);