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) = C22×C4 |
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 | |
415 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 415 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,24,21,25,28,23)(19,22,27,31,26,30,32,29);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(9,11,12,16)(10,14,13,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(2,6)(5,7)(10,13)(14,15)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=fg,b^2=eg,c^2=g,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29340, #6461415, #6461421, … |
| maximal normal subgroup of | C2×#29340, #6461415, #6461421, … |
| maximal quotient of | #362806, #364360, #368479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22328);