Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | Aut(G) order 4096 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
512 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 512 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:
(2,9)(3,14)(4,19)(5,15)(10,28)(11,30)(12,21)(13,27)(16,20)(17,18)(22,23)(24,32)(25,26)(29,31);(1,2)(3,10)(4,11)(5,21)(6,24)(7,12)(8,13)(9,15)(14,30)(16,31)(17,26)(18,29)(19,25)(20,28)(22,27)(23,32);(1,3,8,18)(2,10,13,29)(4,15,19,22)(5,20,23,16)(6,14,7,17)(9,25,27,11)(12,26,24,30)(21,28,32,31);(1,4,7,16)(2,11,12,31)(3,15,17,5)(6,20,8,19)(9,26,21,10)(13,25,24,28)(14,23,18,22)(27,30,32,29)Polycyclic presentation:
2^8|c^2=h,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #484483, #484487, #484495, … |
| maximal normal subgroup of | #484483, #484487, #484495, … |
| maximal quotient of | #168524, #168534, #168584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11286);