Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,23,27)(19,24,30,39)(20,31,33,41)(21,26,22,35)(25,36,42,32)(28,43,45,48)(29,38,34,40)(37,44,47,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,40)(27,39)(31,43)(32,46)(33,45)(35,38)(36,44)(41,48)(42,47);(17,20,23,33)(18,25,27,42)(19,28,30,45)(21,32,22,36)(24,37,39,47)(26,41,35,31)(29,44,34,46)(38,43,40,48);(3,10)(6,9)(11,16)(14,15)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,39,40)(25,41,42,31)(28,44,45,46)(37,43,47,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,30)(24,39)(28,45)(29,34)(37,47)(38,40)(43,48)(44,46)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62210, #10579475, #10579522, … |
| maximal normal subgroup of | C2×#62210, #10579475, #10579522, … |
| maximal quotient of | #6386455, #6388582, #6388590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55198);