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 | #3558 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
475 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 475 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:
(1,2,6,10)(3,11,13,15)(4,8,5,9)(7,14,16,12)(17,18,21,24)(19,25,26,32)(20,28,30,23)(22,29,31,27);(1,3)(2,7)(4,14)(5,12)(6,13)(8,11)(9,15)(10,16)(17,19)(18,22)(20,29)(21,26)(23,32)(24,31)(25,28)(27,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(20,30)(23,28)(25,32)(27,29);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,16,11)(19,26)(22,31)(25,32)(27,29)Polycyclic presentation:
2^8|a^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 | #244548, #244549, #244551, … |
| maximal normal subgroup of | #244548, #244549, #244551, … |
| maximal quotient of | #166326, #166344, #166382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12031);