Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,24)(13,18)(14,28)(15,19)(17,26)(20,36)(21,31)(22,32)(23,38)(25,40)(27,34)(29,35)(30,37)(33,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,25)(13,21)(14,23)(15,22)(17,33)(18,31)(19,32)(20,27)(24,40)(26,39)(28,38)(29,37)(30,35)(34,36);(9,12)(10,17)(11,20)(13,26)(14,29)(15,27)(16,30)(18,34)(19,35)(21,37)(22,25)(23,39)(24,28)(31,40)(32,33)(36,38);(2,5)(6,8)(9,13,15,14)(10,18,19,28)(11,21,22,23)(12,26,27,29)(16,31,32,38)(17,34,35,24)(20,37,25,39)(30,40,33,36);(1,3,4,7)(2,6,5,8)(11,22)(16,32)(20,25)(21,23)(30,33)(31,38)(36,40)(37,39)Polycyclic presentation:
2^8|d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #221312, #221322, #221332, … |
| maximal normal subgroup of | #221312, #221322, #221332, … |
| maximal quotient of | #6379564, #6379565, #6380394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54480);