Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,17,14,12,22,16)(11,15,20,18,21,23,19,24);(1,2,3,4)(9,11)(10,15)(12,23)(13,20)(14,21)(16,24)(17,18)(19,22);(6,8)(9,12)(10,14)(11,18)(13,16)(15,19)(17,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=d,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,f^b=fh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11068, #357737, #357738, … |
| maximal normal subgroup of | C2×#11068, #357737, #357738, … |
| maximal quotient of | #115715, #115716, #115763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4056);