Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 131072 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(9,10,14,11)(12,15,13,16)(18,20)(19,25)(24,28)(27,31);(3,7)(4,6)(9,11,14,10)(12,15,13,16)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30);(1,3,5,7)(2,6,8,4)(9,12,14,13)(10,15,11,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31);(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,4,5,6)(2,7,8,3)(9,13,14,12)(10,16,11,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)Polycyclic presentation:
2^8|a^2=h,b^2=gh,c^2=h,d^2=f,e^2=fh,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 | #98333, #253021, #253024, … |
| maximal normal subgroup of | #98333, #253021, #253024, … |
| maximal quotient of | #521476, #522130, #522638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32722);