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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,14,15)(11,13,12,16)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32);(1,2)(3,4)(5,6)(7,8)(10,15)(11,12)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32);(2,6)(3,7)(9,11,14,12)(10,16,15,13)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32);(1,3,5,7)(2,4,6,8)(9,12,14,11)(10,13,15,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,15)(14,16)(17,22,23,31)(18,24,25,30)(19,27,20,28)(21,29,26,32)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=f,d^2=h,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×#58091, #8104158, #8104425, … |
| maximal normal subgroup of | C2×#58091, #8104158, #8104425, … |
| maximal quotient of | #585595, #586347, #586831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51079);