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 | #3197 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,22,26)(19,27,28,40)(20,24,31,38)(21,33,35,25)(23,34,36,30)(29,41,42,48)(32,45,44,39)(37,46,47,43);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(19,28)(23,36)(27,40)(29,42)(30,34)(37,47)(41,48)(43,46);(3,10)(6,9)(11,16)(13,15)(17,19,22,28)(18,23,26,36)(20,29,31,42)(21,34,35,30)(24,37,38,47)(25,40,33,27)(32,46,44,43)(39,48,45,41);(2,7)(6,9)(8,14)(13,15)(17,20)(18,24)(19,29)(21,32)(22,31)(23,37)(25,39)(26,38)(27,41)(28,42)(30,43)(33,45)(34,46)(35,44)(36,47)(40,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,31)(24,38)(29,42)(32,44)(37,47)(39,45)(41,48)(43,46)Polycyclic presentation:
2^8|a^2=h,c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cfh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #485589, C2×#61005, #10575853, … |
| maximal normal subgroup of | #485589, C2×#61005, #10575853, … |
| maximal quotient of | #6374690, #6375045, #6376041, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53993);