Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C25 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
577 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6)(7,10)(8,15,17,11)(9,12)(13,20,16,18)(14,19);(1,2,3,4)(5,7)(6,10)(8,16,17,13)(9,14)(11,18,15,20)(12,19);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20)Polycyclic presentation:
2^8|a^2=g,b^2=d,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11089, #357766, C4×#2032, … |
| maximal normal subgroup of | C2×#11089, #357766, C4×#2032, … |
| maximal quotient of | #115977, #115995, #115999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4077);