Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | #2812 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
915 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18,22,23)(19,26,24,20)(21,29,30,25)(27,28,31,32);(1,3,4,8)(2,5,7,6)(10,13)(11,15)(18,23)(19,24)(21,30)(28,32);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,20,23,26)(21,27,30,31)(25,28,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,29)(30,32);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130112, #485621, #485701, … |
| maximal normal subgroup of | #130112, #485621, #485701, … |
| maximal quotient of | #557018, #557374, #557393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38449);