Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| 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 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,28,23,24)(20,31,34,39)(21,35,37,26)(25,32,41,29)(30,38,36,42)(33,46,47,44)(40,43,48,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,36,37,30)(25,39,41,31)(26,42,35,38)(33,43,47,45)(40,44,48,46);(3,10)(6,9)(11,16)(13,15)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,37,47)(24,31,28,39)(26,40,35,48)(30,43,36,45)(38,44,42,46);(2,7)(6,9)(8,14)(13,15)(17,21)(18,26)(19,30)(20,33)(22,37)(23,36)(24,38)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(39,46)(41,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(21,37)(26,35)(30,36)(33,47)(38,42)(40,48)(43,45)(44,46)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131644, #485030, #485036, … |
| maximal normal subgroup of | #131644, #485030, #485036, … |
| maximal quotient of | #6374692, #6375044, #6376031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54001);