Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1021 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)(9,10,15,19)(11,16,21,30)(12,23,25,33)(13,18,14,27)(17,28,34,24)(20,35,36,40)(22,31,26,32)(29,37,39,38);(1,3)(2,5)(4,8)(6,7)(9,11,15,21)(10,16,19,30)(12,20,25,36)(13,26,14,22)(17,29,34,39)(18,32,27,31)(23,35,33,40)(24,38,28,37);(9,12)(10,17)(11,20)(13,24)(14,28)(15,25)(16,29)(18,33)(19,34)(21,36)(22,37)(23,27)(26,38)(30,39)(31,35)(32,40);(3,8)(5,7)(11,21)(16,30)(20,36)(22,26)(29,39)(31,32)(35,40)(37,38);(2,6)(5,7)(9,13,15,14)(10,18,19,27)(11,22,21,26)(12,24,25,28)(16,31,30,32)(17,33,34,23)(20,37,36,38)(29,35,39,40)Polycyclic presentation:
2^8|a^2=h,b^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61642, #10577245, #10577319, … |
| maximal normal subgroup of | C2×#61642, #10577245, #10577319, … |
| maximal quotient of | #6386046, #6387647, #6387650, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54630);