Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
735 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)(3,5)(4,7)(6,8)(9,10,14,15)(11,13,12,16)(17,18,22,25)(19,26,23,20)(21,24,30,29)(27,32,31,28);(2,3)(4,8)(10,15)(11,12)(17,18,22,25)(19,20,23,26)(21,29,30,24)(27,32,31,28);(2,3)(6,7)(9,11,14,12)(10,16,15,13)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,15,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,15)(14,16)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#56904, #8099632, #8100313, … |
| maximal normal subgroup of | C2×#56904, #8099632, #8100313, … |
| maximal quotient of | #585365, #585999, #586887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49892);