Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | #3221 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
823 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,29)(20,33)(21,28)(22,26)(23,27)(24,32)(25,34)(30,44)(31,39)(35,42)(36,41)(37,40)(38,45)(43,47)(46,48);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,24)(20,30)(21,36)(22,32)(23,31)(25,38)(26,29)(27,41)(28,39)(33,48)(34,47)(35,46)(37,45)(40,43)(42,44);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,22,35)(18,25,26,40)(19,30,32,46)(21,34,23,37)(24,38,29,43)(27,33,28,42)(31,45,36,47)(39,44,41,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(18,26)(24,29)(25,40)(27,28)(33,42)(38,43)(39,41)(44,48);(2,10)(7,11)(8,9)(15,16)(17,21,22,23)(18,27,26,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)Polycyclic presentation:
2^8|b^2=fh,c^2=f,d^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #152147, C2×#61946, #10578625, … |
| maximal normal subgroup of | #152147, C2×#61946, #10578625, … |
| maximal quotient of | #6386341, #6388241, #6388257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54934);