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 | |
659 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)(10,13)(14,16)(18,21)(19,23);(6,8)(10,13)(14,16)(17,18,20,21)(19,22,23,24);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=ef,d^2=efh,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#14216, #514706, #514720, … |
| maximal normal subgroup of | C2×#14216, #514706, #514720, … |
| maximal quotient of | #153805, #153815, #153822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7204);