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) = C27×C4 |
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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(14,16)(18,19);(6,9)(10,12)(13,14,17,16)(15,19,20,18);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18);(5,6)(7,10)(8,9)(11,12)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^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×#38362, #7635874, #7636307, … |
| maximal normal subgroup of | C2×#38362, #7635874, #7636307, … |
| maximal quotient of | #521748, #522356, #522958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31350);