Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,26)(19,27,28,23)(20,24,30,32)(21,25,31,29);(1,3,5,8)(2,6,4,7)(10,13)(11,16)(19,28)(21,31)(23,27)(25,29);(3,8)(6,7)(9,10,12,13)(11,14,16,15)(17,19,22,28)(18,23,26,27)(20,29,30,25)(21,24,31,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,25)(21,23)(22,30)(26,32)(27,31)(28,29);(1,4,5,2)(3,6,8,7)(9,11,12,16)(10,14,13,15)(17,21)(18,25)(19,24)(20,23)(22,31)(26,29)(27,30)(28,32)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133807, #251966, #251992, … |
| maximal normal subgroup of | #133807, #251966, #251992, … |
| maximal quotient of | #576668, #576670, #576774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40212);