Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
533 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 533 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,21,23,24)(25,26,28,30)(27,29,31,32);(2,5)(6,8)(10,11)(14,15)(19,23)(21,24)(25,26,28,30)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,23)(22,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,18)(19,21)(20,22)(23,24)(25,27,28,31)(26,29,30,32)Polycyclic presentation:
2^8|a^2=efh,b^2=ef,c^2=e,d^2=gh,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | #15210⋊C3 |
| maximal subgroup of | C2×#15210, #608513, #608519, … |
| maximal normal subgroup of | C2×#15210, #608513, #608519, … |
| maximal quotient of | #154182, #154411, #154430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8198);