Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 | #3524 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ 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:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,28)(20,32)(21,26)(22,27)(23,36)(24,31)(25,33)(29,44)(30,40)(34,41)(35,39)(37,42)(38,45)(43,47)(46,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,29)(21,35)(22,31)(23,30)(25,38)(26,40)(27,28)(32,48)(33,47)(34,46)(36,39)(37,45)(41,44)(42,43);(17,20,22,34)(18,25,27,42)(19,29,31,46)(21,33,23,37)(24,38,28,43)(26,41,36,32)(30,45,35,47)(39,48,40,44);(3,10)(6,9)(11,16)(14,15)(17,21,22,23)(18,26,27,36)(19,30,31,35)(20,33,34,37)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,27)(19,31)(25,42)(26,36)(29,46)(30,35)(32,41)(45,47)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62174, #10579425, #10579522, … |
| maximal normal subgroup of | C2×#62174, #10579425, #10579522, … |
| maximal quotient of | #6386437, #6388482, #6388621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55162);