Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10512 |
| automorphism | Aut(G) = C22×#72231160 (order 4096) · Inn(G) = #394 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| 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 | |
183 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9)(3,10,11,7)(4,6,13,8)(12,15,16,14)(17,18)(19,27)(20,25)(21,24)(22,26)(23,29)(28,32)(30,31);(2,6)(3,11)(4,13)(7,15)(8,9)(10,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,32)(25,27)(26,31);(1,3,4,12,5,11,13,16)(2,7,8,15,9,10,6,14)(18,24)(19,28)(21,22)(25,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=g,c^2=dh,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,f^b=fh,e^c=egh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10515, #238808, #238810, … |
| maximal normal subgroup of | C2×#10515, #238808, #238810, … |
| maximal quotient of | #104860, #104865, #104867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3503);