Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| 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 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,9)(10,12)(13,14,18,20)(15,16,19,17);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,15,20,19);(1,2)(3,4)(13,17,18,16)(14,19,20,15);(5,6)(7,10)(8,9)(11,12)Polycyclic presentation:
2^8|a^2=g,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#59132, #7666571, #7666664, … |
| maximal normal subgroup of | C2×#59132, #7666571, #7666664, … |
| maximal quotient of | #531693, #531725, #531862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 52120);