Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60771 |
| 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 | #3512 | 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)(17,18,22,27)(19,23,28,37)(20,25,32,41)(21,24,34,26)(29,36,31,40)(30,38,43,35)(33,39,46,42)(44,48,45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,31)(21,30)(22,28)(24,38)(25,40)(26,35)(27,37)(29,32)(33,45)(34,43)(36,41)(39,47)(42,48)(44,46);(3,10)(6,9)(11,16)(13,15)(18,24)(19,28)(21,34)(23,35)(25,39)(26,27)(29,31)(33,46)(36,47)(37,38)(40,48)(41,42);(2,7)(6,9)(8,14)(13,15)(17,20,22,32)(18,25,27,41)(19,29,28,31)(21,33,34,46)(23,36,37,40)(24,39,26,42)(30,44,43,45)(35,47,38,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,32)(25,41)(29,31)(33,46)(36,40)(39,42)(44,45)(47,48)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,d^b=dh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6352006, #6352008, #6352010, … |
| maximal normal subgroup of | #6352006, #6352008, #6352010, … |
| maximal quotient of | #6374697, #6374699, #6375046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54008);