Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | Aut(G) = C2×#72216396 (order 2048) · Inn(G) = #2404 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2404 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 698 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) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16);(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)Polycyclic presentation:
2^8|b^2=e,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #511853, #511854, #511856, … |
| maximal normal subgroup of | #511853, #511854, #511856, … |
| maximal quotient of | #251827, #251828, #251829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16984);