Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1437 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,7,10,6,11)(3,8,15,23,14,24,20,22)(4,16,18,9,17,25,19,26)(13,27,28,21,29,30,31,32);(1,3,6,15,7,14,5,20)(2,8,12,22,10,24,11,23)(4,13,19,28,17,29,18,31)(9,21,25,27,26,32,16,30);(1,4,7,17)(2,9)(3,13,14,29)(5,18,6,19)(8,21)(10,26)(11,16)(12,25)(15,28,20,31)(22,27)(23,30)(24,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10955, #354322, #355561, … |
| maximal normal subgroup of | C2×#10955, #354322, #355561, … |
| maximal quotient of | #110136, #110137, #110140, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3943);