Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6,9,10)(7,11,12,8);(6,10)(8,11)(14,18)(16,19);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18);(5,7)(6,8)(9,12)(10,11)(13,15,17,20)(14,19,18,16);(5,8)(6,7)(9,11)(10,12)(13,16,17,19)(14,20,18,15)Polycyclic presentation:
2^8|a^2=fg,d^2=h,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 | C2×#39869, #7637518, #7638118, … |
| maximal normal subgroup of | C2×#39869, #7637518, #7638118, … |
| maximal quotient of | #521903, #522463, #522985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32857);