Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | Aut(G) order 4096 |
| 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 | |
552 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 552 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:
(2,8)(3,13)(4,6)(5,16)(9,24)(10,17)(11,22)(12,23)(14,29)(15,20)(18,19)(21,25)(26,32)(27,30);(1,2)(3,9)(4,10)(5,17)(6,11)(7,12)(8,16)(13,25)(14,30)(15,27)(18,23)(19,22)(20,26)(21,28)(24,29)(31,32);(1,3)(2,9)(4,13)(5,14)(6,20)(7,15)(8,21)(10,25)(11,26)(12,27)(16,28)(17,30)(18,31)(19,29)(22,24)(23,32);(1,4,7,6)(2,10,12,11)(3,13,15,20)(5,16,19,18)(8,22,23,17)(9,25,27,26)(14,28,29,31)(21,24,32,30)Polycyclic presentation:
2^8|d^2=h,e^2=gh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gh,e^b=eg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #495737, #498126, #499785, … |
| maximal normal subgroup of | #495737, #498126, #499785, … |
| maximal quotient of | #484215, #484217, #484219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26529);