Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12725 |
| automorphism | Aut(G) = C2×#72226369 (order 2048) · Inn(G) = #1782 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1782 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | #534 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 69 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)(3,11)(4,15)(5,18,19,9)(6,10)(7,13,21,28)(8,17)(12,25)(14,24)(16,22,31,30)(20,32,29,26)(23,27);(1,3,6,14)(2,7,10,21)(4,12,17,27)(5,13,19,28)(8,20,15,29)(9,11,18,24)(16,26,31,32)(22,23,30,25);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,20,21,29)(9,22,18,30)(11,23,24,25)(13,26)(14,27)(19,31)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=g,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fg,e^c=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94454, C2×#12726, #493457, … |
| maximal normal subgroup of | #94454, C2×#12726, #493457, … |
| maximal quotient of | #129290, #129293, #129376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5714);