Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| 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 | #3204 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
834 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)(19,23)(20,30)(21,33)(22,25)(24,31)(26,35)(27,41)(28,37)(29,45)(32,40)(34,39)(36,43)(38,44)(42,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,37)(26,38)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(3,10)(6,9)(11,16)(13,15)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(26,40,33,30)(29,46,44,43)(38,48,45,41);(2,7)(6,9)(8,14)(13,15)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(18,25)(23,37)(24,39)(26,33)(30,40)(36,47)(38,45)(41,48)Polycyclic presentation:
2^8|c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eh,g^a=gh,c^b=cf,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #485591, C2×#60975, #10575819, … |
| maximal normal subgroup of | #485591, C2×#60975, #10575819, … |
| maximal quotient of | #6374682, #6375052, #6375991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53963);