Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22)(23,30)(24,28)(25,33)(26,29)(27,31)(32,35)(34,36);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,23,26,32)(22,27,29,36)(24,31,25,34)(28,35,33,30);(8,16)(11,18)(14,20)(17,19)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^c=de,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #166748, #166750, #166780, … |
| maximal normal subgroup of | #166748, #166750, #166780, … |
| maximal quotient of | #327778, #327786, #327794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22633);