Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(1,2)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,23)(21,30)(22,28)(25,26)(27,29)(31,32);(10,13)(14,16)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)Polycyclic presentation:
2^8|a^2=h,b^2=egh,c^2=h,d^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #123178, #123188, C2×#29550, … |
| maximal normal subgroup of | #123178, #123188, C2×#29550, … |
| maximal quotient of | #364013, #370863, #370945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22538);