Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,5)(6,7)(10,14)(12,16)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30);(1,3,6,4)(2,5,7,8)(9,10)(11,12)(13,14)(15,16)(18,20)(23,25)(24,28)(30,31);(2,7)(3,4)(10,14)(11,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,4,6,3)(2,5,7,8)(9,11,13,15)(10,12,14,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,5)(2,3)(4,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=fh,d^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132146, #253434, #253480, … |
| maximal normal subgroup of | #132146, #253434, #253480, … |
| maximal quotient of | #560424, #560625, #561431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43175);