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) = 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 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
493 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)(9,10,13,15,14,18,12,17)(11,16,21,22,20,23,19,24);(5,7,8,6)(10,15)(11,19)(12,13)(17,18)(20,21)(22,24);(1,2,3,4)(11,20)(16,23)(19,21)(22,24);(9,11,12,19,14,20,13,21)(10,16,17,24,18,23,15,22)Polycyclic presentation:
2^8|a^2=g,b^2=eg,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23192, #6382766, #6382774, … |
| maximal normal subgroup of | C2×#23192, #6382766, #6382774, … |
| maximal quotient of | #231416, #231433, #231436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16180);