Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18)(11,20,21,15)(12,24,25,17)(14,16,27,19)(22,29,34,33)(23,28,35,30)(26,32,37,31)(36,39,40,38);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,22)(13,21)(14,23)(16,30)(17,29)(18,20)(19,28)(24,33)(25,34)(26,36)(27,35)(31,39)(32,38)(37,40);(3,8)(5,7)(10,16)(11,21)(14,27)(15,28)(17,31)(18,19)(20,30)(22,34)(24,32)(26,37)(29,38)(33,39);(2,6)(5,7)(9,12)(10,17)(11,22)(13,25)(14,26)(15,29)(16,31)(18,24)(19,32)(20,33)(21,34)(23,36)(27,37)(28,38)(30,39)(35,40);(11,21)(12,25)(15,20)(17,24)(23,35)(26,37)(28,30)(31,32)Polycyclic presentation:
2^8|a^2=h,b^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6671645, #6671646, #6671657, … |
| maximal normal subgroup of | #6671645, #6671646, #6671657, … |
| maximal quotient of | #6374288, #6374461, #6374991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53823);