Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = C22×#7622880 (order 2048) · Inn(G) = #535 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2404 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,7)(4,8)(5,10)(6,11)(9,13)(12,23)(14,21)(15,32)(16,24)(17,25)(18,27)(19,28)(20,29)(22,31)(26,30);(1,3)(2,7)(4,15,16,31)(5,14)(6,13)(8,22,24,32)(9,11)(10,21)(12,17,29,18)(19,30)(20,25,23,27)(26,28);(1,4)(2,8)(3,12)(5,18)(6,16)(7,20)(9,23)(10,27)(11,24)(13,29)(14,31)(15,30)(17,19)(21,32)(22,26)(25,28);(1,3)(2,9)(4,12)(5,14)(6,13)(7,11)(8,23)(10,26)(15,17)(16,29)(18,31)(19,30)(20,24)(21,28)(22,27)(25,32)Polycyclic presentation:
2^8|b^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^b=fh,e^c=eh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #511841, #511842, #511849, … |
| maximal normal subgroup of | #511841, #511842, #511849, … |
| maximal quotient of | #251800, #251801, #251802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16983);