Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | Aut(G) = C23×#7670203 (order 4096) · Inn(G) = C2×#120 · Out(G) = C26 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
778 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 778 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16);(13,16);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|d^2=e,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #248457, #248460, #248461, … |
| maximal normal subgroup of | #248457, #248460, #248461, … |
| maximal quotient of | #257971, #257972, #257975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17128);