Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | Aut(G) = #72286925 (order 4096) · Inn(G) = #1408 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1408 |
| commutator | C4×C8 | C2×C4 |
| Frattini, cosocle | #494 | C22 |
| prime radical | #493 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
73 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,10,7,12,20,27)(3,9,14,25,16,23,30,32)(5,11,19,28,6,21,18,26)(8,17,22,15,24,29,31,13);(2,8)(3,13)(5,6)(9,11)(10,25)(12,24)(14,17)(15,16)(18,19)(21,23)(22,26)(27,32)(28,31)(29,30)Polycyclic presentation:
2^8|a^2=d,c^2=fg,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=cfgh,d^b=def,e^b=egh,f^b=fh,d^c=dfg,e^d=egh,f^d=fh25 conjugacy classes — view the full character table.
| maximal subgroup of | #94820, #94827, #94828, … |
| maximal normal subgroup of | #94820, #94827, #94828, … |
| maximal quotient of | #94567, #94585, #94644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 519);