Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | |
889 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,14,19)(11,21,24,16)(12,26,28,34)(13,20,15,18)(17,29,33,27)(22,36,38,40)(23,31,25,32)(30,37,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,24)(10,16,19,21)(12,22,28,38)(13,23,15,25)(17,30,33,35)(18,31,20,32)(26,40,34,36)(27,37,29,39);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,26)(19,33)(20,34)(21,35)(23,37)(24,38)(25,39)(31,40)(32,36);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,25)(12,27,28,29)(16,31,21,32)(17,26,33,34)(22,37,38,39)(30,40,35,36);(1,3,4,7)(2,6,5,8)(10,19)(11,24)(17,33)(18,20)(22,38)(23,25)(26,34)(37,39)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250348, #250353, #250380, … |
| maximal normal subgroup of | #250348, #250353, #250380, … |
| maximal quotient of | #6386471, #6388492, #6388610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55078);