Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
491 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)(13,14,17,21,16,20,25,19)(15,22,24,27,23,18,26,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,23)(17,24)(19,27)(20,22)(21,28)(25,26);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(17,25)(18,27)(20,21)(22,28)(24,26);(6,9)(10,12)(14,20)(15,23)(19,21)(24,26)Polycyclic presentation:
2^8|a^2=f,c^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24564, #6440564, #6440570, … |
| maximal normal subgroup of | C2×#24564, #6440564, #6440570, … |
| maximal quotient of | #274975, #274977, #274983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17552);