Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | Aut(G) = #72311826 (order 2048) · Inn(G) = #1999 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 57 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 — 3 generators on 32 points:
(1,2)(3,14)(4,18)(5,13)(6,11)(7,12)(8,15)(9,16)(10,22)(17,30)(19,31)(20,26)(21,27)(23,29)(24,28)(25,32);(1,3)(2,8)(4,19)(5,16)(6,24)(7,17)(9,27)(10,28)(11,25)(12,26)(13,31)(14,18)(15,22)(20,21)(23,32)(29,30);(1,3)(2,9)(4,20)(5,24)(6,16)(7,17)(8,10)(11,30)(12,28)(13,14)(15,23)(18,25)(19,21)(22,32)(26,27)(29,31)Polycyclic presentation:
2^8|d^2=gh,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=cf,d^b=dg,g^b=gh,d^c=dh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13318, #498825, #499269, … |
| maximal normal subgroup of | C2×#13318, #498825, #499269, … |
| maximal quotient of | #142242, #142243, #142244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6306);