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 | #525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
837 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)(17,18,23,27)(19,24,30,40)(20,31,33,41)(21,26,22,35)(25,36,42,32)(28,43,46,48)(29,38,34,39)(37,44,47,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,39)(27,40)(31,43)(32,45)(33,46)(35,38)(36,44)(41,48)(42,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,23,33)(18,25,27,42)(19,28,30,46)(21,32,22,36)(24,37,40,47)(26,41,35,31)(29,44,34,45)(38,43,39,48);(3,11)(6,9)(10,16)(13,15);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,40,39)(25,41,42,31)(28,44,46,45)(37,43,47,48)Polycyclic presentation:
2^8|a^2=h,c^2=fh,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 | #249901, #249906, #249933, … |
| maximal normal subgroup of | #249901, #249906, #249933, … |
| maximal quotient of | #6386171, #6387835, #6387846, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54745);