Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| 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 | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,8,7)(11,15)(13,16)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32);(1,3)(2,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(18,23)(19,25)(24,30)(26,31);(1,2,5,4)(3,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,4,5,2)(3,6,8,7)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fh,e^2=h,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 | #130597, #254346, #254369, … |
| maximal normal subgroup of | #130597, #254346, #254369, … |
| maximal quotient of | #521410, #522178, #522584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32461);