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) = C28 |
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 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6)(7,8)(9,10)(11,12);(6,10)(8,11)(13,14,17,20)(15,19,16,18);(1,2)(3,4)(13,15,17,16)(14,18,20,19);(5,7)(6,11)(8,10)(9,12);(5,8,9,11)(6,7,10,12)(13,16,17,15)(14,19,20,18)Polycyclic presentation:
2^8|b^2=h,c^2=h,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 | #130923, #248546, #248547, … |
| maximal normal subgroup of | #130923, #248546, #248547, … |
| maximal quotient of | #521466, #522177, #522727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32580);