Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,13)(11,15)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#52566, #8092280, #8092470, … |
| maximal normal subgroup of | C2×#52566, #8092280, #8092470, … |
| maximal quotient of | #582333, #582520, #582550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45554);