Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,17,21,18,22,16,20)(15,23,25,19,26,27,24,28);(1,2)(3,4)(6,9)(10,11)(13,15,18,26)(14,19,22,28)(16,25,17,24)(20,27,21,23);(2,4)(14,20)(16,17)(19,27)(21,22)(23,28)(24,25);(5,7,8,12)(6,10,9,11)(14,21)(16,17)(19,23)(20,22)(24,25)(27,28)Polycyclic presentation:
2^8|a^2=eh,b^2=h,d^2=ef,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh,e^d=eh,f^d=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26900, #6439640, #6439661, … |
| maximal normal subgroup of | C2×#26900, #6439640, #6439661, … |
| maximal quotient of | #312036, #312038, #312064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19888);