Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
939 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,18)(14,29)(15,19)(17,27)(20,35)(21,31)(22,32)(23,38)(25,39)(26,34)(28,40)(30,37)(33,36);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,23)(15,21)(17,33)(18,32)(19,31)(20,28)(24,39)(26,37)(27,36)(29,38)(30,34)(35,40);(9,12,14,27,15,28,13,26)(10,17,18,24,19,34,29,40)(11,20,23,37,21,25,22,36)(16,30,32,35,31,33,38,39);(1,3,4,7)(2,6,5,8)(11,21)(16,31)(20,25)(22,23)(30,33)(32,38)(35,39)(36,37);(9,13,15,14)(10,18,19,29)(11,22,21,23)(12,26,28,27)(16,32,31,38)(17,24,34,40)(20,36,25,37)(30,35,33,39)Polycyclic presentation:
2^8|c^2=g,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129558, #129909, #129939, … |
| maximal normal subgroup of | #129558, #129909, #129939, … |
| maximal quotient of | #6371177, #6372105, #6372113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53673);