Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) = #72287717 (order 8192) · Inn(G) = C2×D16 · Out(G) = C4×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×D32 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,8,12)(7,11,18,24)(9,22,20,36)(10,23,21,15)(13,29,27,16)(14,31,28,17)(19,32,33,26)(25,35,30,34);(6,11)(7,16)(9,17)(10,21)(12,24)(13,14)(15,32)(18,29)(19,35)(20,31)(22,25)(23,26)(27,28)(30,36)(33,34);(1,3,4,2)(6,12)(11,24)(13,27)(14,28)(15,23)(22,36)(25,30)(26,32)Polycyclic presentation:
2^8|a^2=e,c^2=eh,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94614, #151085, #151115, … |
| maximal normal subgroup of | #94614, #151085, #151115, … |
| maximal quotient of | #151080, #151111, #151126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6642);