Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31);(1,2,5,7)(3,4,8,6)(18,23)(20,25)(24,29)(28,31);(1,3,5,8)(2,6,7,4)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(1,4)(2,3)(5,6)(7,8)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^2=fg,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 | C2×#47595, #7701554, #7701594, … |
| maximal normal subgroup of | C2×#47595, #7701554, #7701594, … |
| maximal quotient of | #576950, #577029, #577038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40583);