Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | Aut(G) = #72231951 (order 2048) · Inn(G) = #2760 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2760 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,15,18,8)(5,9)(6,10)(12,25)(13,21,27,23)(16,20)(17,28,30,22)(19,24)(26,31)(29,32);(1,3)(2,7)(4,16)(5,13)(6,14)(8,20)(9,23)(10,11)(12,18)(15,25)(17,29)(19,27)(21,24)(22,32)(26,30)(28,31);(1,4)(2,8)(3,12,14,16)(5,17)(6,18)(7,20,11,25)(9,22)(10,15)(13,26,27,29)(19,30)(21,31,23,32)(24,28);(1,5,6,19)(2,9,10,24)(3,13,14,27)(4,17,18,30)(7,21,11,23)(8,22,15,28)(12,26,16,29)(20,31,25,32)Polycyclic presentation:
2^8|a^2=fg,c^2=ef,d^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,f^b=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151344, #151345, #151460, … |
| maximal normal subgroup of | #151344, #151345, #151460, … |
| maximal quotient of | #346224, #346227, #346229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25689);