Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18,20,23)(19,25,27,32)(21,30,29,24)(22,31,26,28);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,26)(20,23)(21,24)(22,27)(25,31)(28,32)(29,30);(11,15)(13,16)(17,19,21,28,20,27,29,31)(18,22,24,25,23,26,30,32);(1,3,4,8)(2,6,5,7)(18,23)(22,26)(24,30)(25,32)Polycyclic presentation:
2^8|a^2=egh,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27901, #6458731, #6458741, … |
| maximal normal subgroup of | C2×#27901, #6458731, #6458741, … |
| maximal quotient of | #360671, #363352, #367286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20889);