Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,7,8)(9,10,14,11,13,17,15,18)(12,20,23,21,22,19,24,16);(1,2,4,3)(9,11,14,17,13,18,15,10)(12,21,23,19,22,16,24,20);(6,8)(9,12,13,22)(10,16,17,21)(11,19,18,20)(14,24,15,23);(1,3,4,2)(10,17)(11,18)(16,21)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28477, #6465628, #6465639, … |
| maximal normal subgroup of | C2×#28477, #6465628, #6465639, … |
| maximal quotient of | #321443, #321449, #321459, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21465);