Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33531 |
| automorphism | Aut(G) order 8192 |
| 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 | |
561 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 561 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:
(2,7)(4,14)(5,17)(8,22)(9,21)(10,25)(12,27)(13,26)(15,19)(16,30)(18,31)(20,23)(24,32)(28,29);(1,2)(3,8)(4,15)(5,19)(6,9)(7,14)(10,17)(11,23)(12,28)(13,29)(16,32)(18,25)(20,30)(21,31)(22,27)(24,26);(1,3)(2,8)(4,16)(5,13)(6,11)(7,20)(9,23)(10,24)(12,18)(14,30)(15,32)(17,26)(19,29)(21,22)(25,28)(27,31);(3,11)(8,23)(12,16)(13,26)(20,22)(24,29)(27,30)(28,32)Polycyclic presentation:
2^8|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 | #142616, #142617, #142620, … |
| maximal normal subgroup of | #142616, #142617, #142620, … |
| maximal quotient of | #484199, #484201, #484546, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26519);