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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
800 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18,22,25)(19,21,24,30)(20,23,27,31)(26,29,28,32);(1,2,4,6)(3,5,8,7)(9,10,13,12)(11,14,16,15)(19,24)(20,27)(21,30)(23,31);(9,11,13,16)(10,14,12,15)(17,19,22,24)(18,21,25,30)(20,28,27,26)(23,32,31,29);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,15,16,14)(17,20)(18,23)(19,26)(21,29)(22,27)(24,28)(25,31)(30,32);(1,3)(2,5)(4,8)(6,7)(17,21)(18,24)(19,25)(20,29)(22,30)(23,28)(26,31)(27,32)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=gh,d^2=g,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 | #133704, #500543, #502834, … |
| maximal normal subgroup of | #133704, #500543, #502834, … |
| maximal quotient of | #561184, #562272, #564935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43317);