Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
443 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 111 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 — 4 generators on 32 points:
(1,2,7,12)(3,13,4,15)(5,10,6,11)(8,16,9,14)(17,18,22,20)(19,25,27,32)(21,29,31,24)(23,30,26,28);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,13,11,15)(17,19,22,27)(18,23,20,26)(21,30,31,28)(24,32,29,25);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,13,11,15)(17,20,22,18)(19,26,27,23)(21,24,31,29)(25,30,32,28);(1,5,7,6)(2,10,12,11)(3,14,4,16)(8,15,9,13)(19,27)(23,26)(25,32)(28,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244552, #244553, #244559, … |
| maximal normal subgroup of | #244552, #244553, #244559, … |
| maximal quotient of | #166332, #166338, #166388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12037);