Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 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)(13,14,18,20,17,22,16,21)(15,19,23,28,24,26,25,27);(1,2)(3,4)(6,9)(10,12)(13,15,16,25,17,24,18,23)(14,19,21,27,22,26,20,28);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(15,23)(16,18)(19,26)(21,22)(24,25);(6,9)(10,12)(15,24)(19,26)(23,25)(27,28)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25244, #6441735, #6441750, … |
| maximal normal subgroup of | C2×#25244, #6441735, #6441750, … |
| maximal quotient of | #285915, #285917, #285923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18232);