Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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×D16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
100 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)(7,11)(8,20)(9,23)(10,12)(13,16)(14,17)(15,21)(18,30)(19,24)(22,36)(25,32)(26,31)(27,33)(28,34)(29,35);(6,11)(7,16)(8,17)(9,21)(12,24)(13,14)(15,30)(18,35)(19,33)(20,25)(22,34)(23,26)(27,28)(29,36)(31,32);(1,3,4,2)(6,12)(11,24)(13,27)(14,28)(15,23)(20,36)(25,29)(26,30)Polycyclic presentation:
2^8|a^2=e,c^2=e,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 | #151083, #151116, #151117, … |
| maximal normal subgroup of | #151083, #151116, #151117, … |
| maximal quotient of | #151078, #151112, #151113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6640);