Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10026 |
| automorphism | Aut(G) = C22×#72231160 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,10,13,11)(4,15,6,8)(7,14,16,12)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,32)(29,31);(1,3,5,13)(2,7,9,16)(4,14,6,12)(8,11,15,10)(22,31)(25,32);(1,2)(3,11)(4,8)(5,9)(6,15)(7,12)(10,13)(14,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,f^b=fg,d^c=dh,e^c=eh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10029, #232676, #232678, … |
| maximal normal subgroup of | C2×#10029, #232676, #232678, … |
| maximal quotient of | #102432, #102438, #102443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3017);