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 | |
517 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 517 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)(4,14)(8,20)(9,18)(10,15)(12,26)(13,25)(16,29)(19,21)(22,32)(23,30)(24,28)(27,31);(1,2)(3,8)(4,15)(5,10)(6,9)(7,14)(11,21)(12,27)(13,22)(16,30)(17,23)(18,29)(19,28)(20,26)(24,32)(25,31);(4,16)(7,18)(12,24)(14,29)(15,30)(19,20)(26,28)(27,32);(1,3,5,13)(2,8,10,22)(4,12,14,28)(6,11,17,25)(7,19,15,27)(9,21,23,31)(16,24,29,26)(18,20,30,32)Polycyclic presentation:
2^8|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 | #484465, #484467, #484473, … |
| maximal normal subgroup of | #484465, #484467, #484473, … |
| maximal quotient of | #168505, #168515, #168567, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11278);