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 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,7)(10,11)(13,14)(15,19)(16,18)(17,20);(2,4)(5,6,9,7)(8,11,12,10)(13,14,16,18)(15,17,20,19);(1,2)(3,4)(13,15)(14,17)(16,20)(18,19);(5,7,9,6)(8,11,12,10)(13,15)(14,17)(16,20)(18,19);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,14)(15,17)(16,18)(19,20)Polycyclic presentation:
2^8|b^2=f,d^2=fg,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×#37722, #7634984, #7635021, … |
| maximal normal subgroup of | C2×#37722, #7634984, #7635021, … |
| maximal quotient of | #521376, #522107, #522618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30710);