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)(13,14,18,21,17,22,16,23)(15,24,25,20,26,19,27,28);(1,2)(3,4)(6,9)(10,12)(13,15)(14,19)(16,27)(17,26)(18,25)(20,23)(21,28)(22,24);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,24)(22,28);(6,9)(10,12)(13,16,17,18)(14,21,22,23)(15,25,26,27)(19,20,24,28)Polycyclic presentation:
2^8|a^2=fh,c^2=eh,d^2=h,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×#24569, #6440564, #6440569, … |
| maximal normal subgroup of | C2×#24569, #6440564, #6440569, … |
| maximal quotient of | #275008, #275010, #275016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17557);