Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,25,17,21)(16,20,27,22)(18,28,26,23);(1,2)(3,4)(6,9)(10,12)(13,15,18,16,19,17,26,27)(14,20,23,25,24,22,28,21);(2,4)(5,7,8,11)(6,10,9,12)(13,16,26,15,19,27,18,17)(14,21,28,22,24,25,23,20);(6,9)(10,12)(13,17,18,27,19,15,26,16)(14,22,23,21,24,20,28,25)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=eh,d^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27260, #6440730, #6440759, … |
| maximal normal subgroup of | C2×#27260, #6440730, #6440759, … |
| maximal quotient of | #313613, #313615, #313621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20248);