Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | Aut(G) = C2×#72236986 (order 4096) · Inn(G) = #2404 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 121 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)(3,8)(4,15)(5,10)(6,21)(7,12)(9,19)(11,24)(13,29)(14,20)(16,32)(17,31)(18,28)(22,26)(23,27)(25,30);(1,3,7,10)(2,8,12,5)(4,16)(6,14,24,27)(9,25)(11,23,21,20)(13,18)(15,29)(17,26)(19,31)(22,30)(28,32);(1,4)(2,9)(3,13)(5,17)(6,22)(7,18)(8,25)(10,16)(11,28)(12,26)(14,31)(15,21)(19,24)(20,29)(23,32)(27,30);(1,5,7,8)(2,10,12,3)(4,17,18,25)(6,23,24,20)(9,16,26,13)(11,14,21,27)(15,30,28,31)(19,29,22,32)Polycyclic presentation:
2^8|b^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152683, #511860, #511861, … |
| maximal normal subgroup of | #152683, #511860, #511861, … |
| maximal quotient of | #251941, #251943, #251944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16989);