Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | Aut(G) = #72314018 (order 4096) · Inn(G) = C2×#447 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 551 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,9)(5,18)(6,16)(7,11)(8,14)(10,21)(13,29)(15,30)(17,20)(19,24)(22,31)(23,27)(25,28)(26,32);(1,3)(2,8)(4,13)(5,19)(6,23)(7,15)(9,24)(10,27)(11,25)(12,18)(14,21)(16,26)(17,28)(20,29)(22,32)(30,31);(1,2)(3,8)(4,16)(5,20)(6,9)(7,11)(10,22)(12,28)(13,26)(14,30)(15,25)(17,18)(19,29)(21,31)(23,24)(27,32);(1,4,7,6)(2,9,11,16)(3,13,15,23)(5,17,22,21)(8,24,25,26)(10,18,20,31)(12,29,30,27)(14,19,28,32)Polycyclic presentation:
2^8|d^2=h,e^2=gh,g^2=h,b^a=be,c^a=cf,e^a=eg,g^a=gh,e^b=eg,f^b=fh,g^b=gh,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #486333, #486334, #486338, … |
| maximal normal subgroup of | #486333, #486334, #486338, … |
| maximal quotient of | #484203, #484204, #484549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26521);