Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,21,23,24);(6,8)(10,11)(14,15)(17,18,20,22)(19,21,23,24);(5,6)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(1,3,4,2)(9,12)(10,14)(11,15)(13,16)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^8|a^2=h,b^2=ef,c^2=e,d^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14180, #514669, #514716, … |
| maximal normal subgroup of | C2×#14180, #514669, #514716, … |
| maximal quotient of | #153792, #153810, #153821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7168);