Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | Aut(G) = C2×D4×#442 (order 1024) · Inn(G) = D4×D8 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,16)(5,6)(9,26)(10,28)(11,30)(12,25)(14,31)(15,21)(17,22)(19,20)(23,27)(24,29);(1,2)(3,9)(4,17)(5,11)(6,22)(7,12)(8,16)(10,14)(13,27)(15,28)(18,23)(19,25)(20,30)(21,24)(26,31)(29,32);(1,3)(2,9)(4,14)(5,21)(6,13)(7,15)(8,23)(10,17)(11,24)(12,28)(16,18)(19,32)(20,31)(22,27)(25,29)(26,30);(1,3)(2,10)(4,18)(5,13)(6,21)(7,15)(8,24)(9,17)(11,23)(12,26)(14,16)(19,31)(20,32)(22,29)(25,27)(28,30)Polycyclic presentation:
2^8|e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gh,d^b=de,e^b=eg,g^b=gh,f^c=fh,e^d=eg,g^d=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#33854, #7623655, #7623662, … |
| maximal normal subgroup of | C2×#33854, #7623655, #7623662, … |
| maximal quotient of | #491712, #491713, #491715, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26842);