Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 | |
1009 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)(11,19)(12,23)(13,15)(14,27)(16,22)(17,24)(18,26)(20,34)(21,32)(25,31)(28,29)(30,40)(33,35)(36,37)(38,39);(1,3,4,8)(2,5,6,7)(10,15)(16,29)(17,23)(18,27)(19,32)(30,37)(31,38)(33,34);(3,8)(5,7)(9,11,13,21)(10,16,15,29)(12,20,24,35)(14,28,26,22)(17,30,23,37)(18,32,27,19)(25,40,39,36)(31,34,38,33);(2,6)(5,7)(9,12)(10,17)(11,20)(13,24)(14,25)(15,23)(16,30)(18,31)(19,33)(21,35)(22,36)(26,39)(27,38)(28,40)(29,37)(32,34);(12,24)(17,23)(20,35)(25,39)(30,37)(31,38)(33,34)(36,40)Polycyclic presentation:
2^8|b^2=fh,c^2=h,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #152122, #249845, #249847, … |
| maximal normal subgroup of | #152122, #249845, #249847, … |
| maximal quotient of | #6374316, #6374468, #6374978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53816);