Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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 | |
888 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)(14,27)(15,19)(17,25)(20,34)(21,30)(22,31)(23,37)(26,32)(28,33)(29,35)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,19,31)(12,20,26,36)(13,21,14,23)(17,29,33,40)(18,30,27,37)(24,34,32,39)(25,35,28,38);(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,32)(19,33)(21,35)(22,36)(23,38)(24,27)(30,39)(31,40)(34,37);(2,5)(6,8)(9,13,15,14)(10,18,19,27)(11,21,22,23)(12,25,26,28)(16,30,31,37)(17,32,33,24)(20,35,36,38)(29,39,40,34);(1,3,4,7)(2,6,5,8)(10,19)(11,22)(17,33)(18,27)(20,36)(21,23)(24,32)(35,38)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=df,e^b=eh,g^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61530, #10576867, #10576931, … |
| maximal normal subgroup of | C2×#61530, #10576867, #10576931, … |
| maximal quotient of | #6379588, #6379596, #6380436, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54518);