Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,9,15,11,16,10,14)(7,17,20,28,21,27,19,12)(8,22,23,30,25,29,24,13)(18,32,34,36,35,26,33,31);(5,7,10,19,11,21,9,20)(6,12,14,27,16,28,15,17)(8,18,24,33,25,35,23,34)(13,26,29,36,30,32,22,31);(1,3,4,2)(5,8,9,23,11,25,10,24)(6,13,15,22,16,30,14,29)(7,18,20,34,21,35,19,33)(12,26,17,31,28,32,27,36)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10902, #353934, #354320, … |
| maximal normal subgroup of | C2×#10902, #353934, #354320, … |
| maximal quotient of | #109524, #109525, #109528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3890);