Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| 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 | |
753 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,14,11)(12,15,13,16)(18,22)(19,25)(24,27)(26,31);(2,5)(3,7)(9,11,14,10)(12,15,13,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,12,14,13)(10,15,11,16);(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,11,15)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=gh,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 | #98407, #260486, #260490, … |
| maximal normal subgroup of | #98407, #260486, #260490, … |
| maximal quotient of | #521501, #522146, #522646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32757);