Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 | #537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
818 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27)(19,23,29,36)(20,25,31,39)(21,24,33,26)(28,35,42,47)(30,34,43,37)(32,38,44,40)(41,45,48,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,29)(18,23,27,36)(20,28,31,42)(21,30,33,43)(24,34,26,37)(25,35,39,47)(32,41,44,48)(38,45,40,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(21,33)(23,34)(25,38)(26,27)(30,43)(32,44)(35,45)(36,37)(39,40)(41,48)(46,47);(2,8)(6,9)(7,16)(14,15)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,38)(26,40)(27,39)(29,42)(30,41)(33,44)(34,45)(36,47)(37,46)(43,48);(19,29)(20,31)(23,36)(25,39)(30,43)(32,44)(34,37)(38,40)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6672293, #6672294, #6672305, … |
| maximal normal subgroup of | #6672293, #6672294, #6672305, … |
| maximal quotient of | #6374373, #6374461, #6374991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53879);