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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
465 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)(15,25,27,21)(16,28,26,20)(17,23,18,22);(1,2)(3,4)(6,9)(10,12)(13,15,18,16,19,27,17,26)(14,20,23,21,24,28,22,25);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,27,19,26,17,15)(14,21,23,28,24,25,22,20);(6,9)(10,12)(13,15,17,26,19,27,18,16)(14,20,22,25,24,28,23,21)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=eh,d^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27277, #6440748, #6440765, … |
| maximal normal subgroup of | C2×#27277, #6440748, #6440765, … |
| maximal quotient of | #314126, #314128, #314134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20265);