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 | #3515 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
817 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)(17,18,22,27)(19,29,32,24)(20,34,36,42)(21,28,23,26)(25,37,41,35)(30,44,46,48)(31,39,33,40)(38,45,43,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,32)(18,24,27,29)(20,30,36,46)(21,31,23,33)(25,38,41,43)(26,39,28,40)(34,48,42,44)(35,45,37,47);(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,34)(27,41)(28,42)(29,43)(31,45)(32,46)(33,47)(39,48)(40,44);(3,10)(6,9)(11,16)(14,15)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,39,29,40)(25,34,41,42)(30,45,46,47)(38,48,43,44);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,27)(19,32)(25,41)(26,28)(30,46)(31,33)(34,42)(45,47)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=h,e^2=fh,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 | #250349, #250352, #250381, … |
| maximal normal subgroup of | #250349, #250352, #250381, … |
| maximal quotient of | #6386471, #6388492, #6388603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55225);