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 | #3540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
848 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,43)(29,38)(30,39)(31,40)(34,41)(36,42)(37,44)(45,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,37)(26,38)(27,40)(32,43)(33,44)(34,46)(35,39)(36,45)(41,47)(42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,41)(27,42)(29,44)(30,45)(31,46)(32,35)(38,47)(39,43)(40,48);(3,11)(6,9)(10,16)(13,15)(17,21,23,22)(18,26,27,35)(19,29,31,30)(20,33,34,36)(24,38,40,39)(25,41,42,32)(28,44,46,45)(37,47,48,43);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,26,27,35)(19,30,31,29)(20,33,34,36)(24,39,40,38)(25,41,42,32)(28,45,46,44)(37,43,48,47)Polycyclic presentation:
2^8|c^2=f,d^2=h,e^2=h,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×#61760, #10577936, #10578011, … |
| maximal normal subgroup of | C2×#61760, #10577936, #10578011, … |
| maximal quotient of | #6386172, #6387836, #6387844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54748);