Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,13,14)(11,15,16,12)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(11,16)(12,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(1,2,4,5)(3,6,7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,15)(14,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#39559, #7636785, #7636858, … |
| maximal normal subgroup of | C2×#39559, #7636785, #7636858, … |
| maximal quotient of | #521416, #522183, #522578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32547);