Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | Aut(G) = #72305983 (order 2048) · Inn(G) = #3109 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #3109 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | #575 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,6,10)(3,11,13,29)(4,16,17,32)(5,21,19,9)(7,22,24,14)(8,26,27,20)(12,28,15,31)(18,23,30,25);(4,17)(8,27)(11,29)(14,22)(16,32)(18,30)(20,26)(28,31);(1,3)(2,7)(4,18)(5,12)(6,13)(8,28)(9,23)(10,24)(11,16)(14,20)(15,19)(17,30)(21,25)(22,26)(27,31)(29,32);(3,12,13,15)(4,17)(7,23,24,25)(8,27)(11,28,29,31)(14,18,22,30)(16,32)(20,26)Polycyclic presentation:
2^8|a^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #143164, #143165, #143168, … |
| maximal normal subgroup of | #143164, #143165, #143168, … |
| maximal quotient of | #494052, #494998, #495481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26677);