Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6)(7,13)(8,18)(9,11)(10,16)(12,19)(14,15)(17,20);(5,7)(6,10)(8,15)(9,17)(11,14)(12,20)(13,18)(16,19);(2,4)(5,6)(7,14)(8,12)(9,11)(10,17)(13,15)(16,20)(18,19);(7,15)(8,19)(10,20)(12,18)(13,14)(16,17);(6,11)(10,14)(12,18)(13,20)Polycyclic presentation:
2^8|f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,e^c=eh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131048, #131050, #131855, … |
| maximal normal subgroup of | #131048, #131050, #131855, … |
| maximal quotient of | #6359112, #6359120, #6359129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54072);