Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| 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 | #3176 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
814 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)(19,27)(20,31)(21,25)(22,35)(23,29)(24,34)(26,32)(28,41)(30,37)(33,46)(36,48)(38,47)(39,44)(40,42)(43,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,29)(18,23,25,27)(20,28,33,43)(22,30,32,42)(24,36,38,39)(26,37,35,40)(31,45,46,41)(34,44,47,48);(17,20,32,47,21,33,22,34)(18,24,35,46,25,38,26,31)(19,28,42,48,29,43,30,44)(23,36,40,41,27,39,37,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,27)(28,43)(30,42)(36,39)(37,40)(41,45)(44,48);(2,8)(6,9)(7,16)(14,15)(18,25)(19,29)(24,38)(26,35)(28,43)(30,42)(31,46)(44,48)Polycyclic presentation:
2^8|b^2=f,c^2=gh,d^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132076, #486088, #7624402, … |
| maximal normal subgroup of | #132076, #486088, #7624402, … |
| maximal quotient of | #6386829, #6389322, #6389353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55365);