Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2847 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 641 conjugacy classes of subgroups — too many to draw. Only its 145 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)(5,18)(9,20)(10,21)(12,28)(13,14)(15,16)(19,23)(22,31)(25,26)(27,29);(1,2,5,7,6,10,18,21)(3,8,13,19,14,24,11,23)(4,9,15,20,17,26,16,25)(12,22,28,30,29,31,27,32);(1,3)(2,8)(4,12)(5,13)(6,14)(7,19)(9,22)(10,24)(11,18)(15,28)(16,27)(17,29)(20,30)(21,23)(25,32)(26,31);(1,4)(2,9)(3,12)(5,15)(6,17)(7,20)(8,22)(10,26)(11,27)(13,28)(14,29)(16,18)(19,30)(21,25)(23,32)(24,31)Polycyclic presentation:
2^8|b^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,f^c=fh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #500797, #500798, #500805, … |
| maximal normal subgroup of | #500797, #500798, #500805, … |
| maximal quotient of | #347878, #347879, #347880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25938);