Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
357 conjugacy classes of subgroups.
| 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)(17,18,23,31,22,27,21,26)(19,24,28,32,20,30,29,25);(1,3,4,8)(2,5,6,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,22)(21,29)(23,28)(25,26)(27,30)(31,32);(10,13)(14,16)(17,20,22,19)(18,25,27,32)(21,29,23,28)(24,31,30,26);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,31)(19,28,20,29)(24,32,30,25)Polycyclic presentation:
2^8|a^2=fh,b^2=egh,c^2=h,d^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×#29571, #6461882, #6461889, … |
| maximal normal subgroup of | C2×#29571, #6461882, #6461889, … |
| maximal quotient of | #363995, #364284, #371288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22559);