Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| 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 | C22×C4 | C24 |
| prime radical | #3188 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
480 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 113 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 — 4 generators on 32 points:
(1,2)(3,13)(4,7)(5,11)(6,20)(8,12)(9,15)(10,22)(14,23)(16,26)(17,29)(18,31)(19,24)(21,25)(27,32)(28,30);(1,3)(2,9)(4,14)(5,15)(6,23)(7,17)(8,18)(10,26)(11,29)(12,28)(13,20)(16,25)(19,31)(21,27)(22,30)(24,32);(1,4,7,2)(3,14,17,9)(5,10,6,19)(8,21,25,12)(11,24,20,22)(13,30,29,32)(15,26,23,31)(16,28,18,27);(1,5,7,6)(2,10,4,19)(3,15,17,23)(8,22,25,24)(9,26,14,31)(11,21,20,12)(13,28,29,27)(16,32,18,30)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244500, #244503, #244507, … |
| maximal normal subgroup of | #244500, #244503, #244507, … |
| maximal quotient of | #169619, #169633, #169684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11410);