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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
848 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)(20,25)(21,24)(22,27)(26,34)(28,36)(29,35)(30,38)(31,40)(32,41)(33,39)(37,44)(42,46)(43,47)(45,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,30)(18,23,27,38)(20,31,32,28)(21,29,34,44)(24,35,26,37)(25,40,41,36)(33,45,46,43)(39,48,42,47);(3,10)(6,9)(11,16)(13,15)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(2,7)(6,9)(8,14)(13,15)(17,20,22,32)(18,25,27,41)(19,28,30,31)(21,33,34,46)(23,36,38,40)(24,39,26,42)(29,43,44,45)(35,47,37,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,32)(25,41)(28,31)(33,46)(36,40)(39,42)(43,45)(47,48)Polycyclic presentation:
2^8|b^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=cf,d^b=dh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131641, #485028, #485034, … |
| maximal normal subgroup of | #131641, #485028, #485034, … |
| maximal quotient of | #6374695, #6375041, #6376036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53997);