Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,12)(14,17)(18,20);(1,3,4,2)(6,10)(8,11);(5,7)(6,11)(8,10)(9,12)(13,14,16,17)(15,18,19,20);(5,8,9,11)(6,7,10,12)(13,15,16,19)(14,18,17,20);(5,7)(6,11)(8,10)(9,12)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=fg,d^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #107600, #182675, #182681, … |
| maximal normal subgroup of | #107600, #182675, #182681, … |
| maximal quotient of | #513558, #513578, #513591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27212);