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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
756 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)(9,10)(11,19)(12,15)(13,22)(14,16)(17,20)(18,23)(21,24);(5,6)(7,8)(9,11,13,12,14,20,23,21)(10,15,18,19,16,24,22,17);(1,2,3,4)(6,8)(10,16)(15,24)(17,19)(18,22);(9,12,13,20,14,21,23,11)(10,17,18,15,16,19,22,24)Polycyclic presentation:
2^8|b^2=e,c^2=fh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25792, #6443224, #6443234, … |
| maximal normal subgroup of | C2×#25792, #6443224, #6443234, … |
| maximal quotient of | #298797, #298802, #298806, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18780);