Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×D32 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,11)(8,12)(9,22)(10,23)(13,19)(14,20)(15,18)(16,24)(17,34)(21,30)(25,31)(26,32)(27,35)(28,33)(29,36);(1,2,3,4)(6,11)(12,24)(13,25)(14,26)(15,23)(22,34)(27,29)(28,30);(6,12)(8,19)(9,20)(10,18)(11,24)(13,14)(15,30)(16,31)(17,32)(21,36)(22,27)(23,28)(25,26)(29,34)(33,35)Polycyclic presentation:
2^8|b^2=dh,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^c=ef,f^c=fg,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13689, #512913, #512914, … |
| maximal normal subgroup of | C2×#13689, #512913, #512914, … |
| maximal quotient of | #151620, #151624, #151632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6677);