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) = C24×C4 |
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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
520 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 520 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:
(2,7)(3,11)(4,15)(8,20)(9,18)(10,16)(13,27)(14,25)(17,26)(19,22)(21,30)(23,32)(24,29)(28,31);(1,2)(3,8)(4,16)(5,10)(6,9)(7,15)(11,22)(12,21)(13,28)(14,23)(17,30)(18,26)(19,29)(20,27)(24,32)(25,31);(4,17)(7,18)(13,24)(15,26)(16,30)(19,20)(27,29)(28,32);(1,3,12,25)(2,8,21,31)(4,13,26,27)(5,14,6,11)(7,19,30,32)(9,22,10,23)(15,29,17,24)(16,28,18,20)Polycyclic presentation:
2^8|d^2=gh,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 | #484465, #484468, #484474, … |
| maximal normal subgroup of | #484465, #484468, #484474, … |
| maximal quotient of | #168544, #168545, #168602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11294);