Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| prime radical | #3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10)(11,17)(12,21)(13,18)(14,23)(15,19)(16,20)(22,24);(1,3)(2,6)(4,8)(5,7)(9,11,15,12)(10,16,19,24)(13,20,14,22)(17,18,21,23);(1,4,6,5)(2,7,3,8)(9,12,15,11)(10,17,19,21)(13,20,14,22)(16,18,24,23);(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,21,24,17)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=h,d^2=g,e^2=h,f^2=gh,b^a=be,c^a=ch,e^a=eg,e^b=eh,f^b=fgh,d^c=dg49 conjugacy classes — view the full character table.
| maximal subgroup of | #221264, #221267, #221278, … |
| maximal normal subgroup of | #221264, #221267, #221278, … |
| maximal quotient of | #160645, #160646, #160649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10753);