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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
814 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)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32);(3,7)(4,6)(9,11,14,10)(12,15,13,16)(18,19)(23,25)(24,26)(30,31);(1,3,5,7)(2,6,8,4)(9,12,14,13)(10,15,11,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,6)(2,7,8,3)(9,13,14,12)(10,16,11,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=h,d^2=f,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 | #6543330, #6543339, #6543345, … |
| maximal normal subgroup of | #6543330, #6543339, #6543345, … |
| maximal quotient of | #521475, #522129, #522637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32719);