Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23895 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
882 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 882 conjugacy classes of subgroups — too many to draw. Only its 141 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)(3,7)(4,8)(5,9)(6,10)(11,17)(12,18)(13,19)(14,27)(15,22)(16,23)(20,25)(21,28)(24,31)(26,30)(29,32);(1,3,6,13)(2,7,10,19)(4,11,15,25)(5,12,16,26)(8,20,22,17)(9,18,23,30)(14,24,28,32)(21,31,27,29);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,24)(13,25)(16,28)(18,29)(19,20)(23,27)(26,32)(30,31);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,23)(11,24)(13,26)(15,28)(17,29)(19,30)(20,31)(22,27)(25,32)Polycyclic presentation:
2^8|b^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,e^c=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139740, #139744, #144362, … |
| maximal normal subgroup of | #139740, #139744, #144362, … |
| maximal quotient of | #244513, #244514, #244515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16884);