Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,3)(6,7)(10,12);(5,6,8,7)(13,14,15,16);(9,10)(11,12)(13,14)(15,16);(1,3,4,2)(5,7,8,6)Polycyclic presentation:
2^8|a^2=h,b^2=eg,d^2=eh,b^a=be,c^a=cf,d^a=de,c^b=cg76 conjugacy classes — view the full character table.
| maximal subgroup of | #136176, #136252, #137321, … |
| maximal normal subgroup of | #136176, #136252, #137321, … |
| maximal quotient of | #153793, #153794, #153818, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7212);