Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #390 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,21)(9,25)(10,14)(11,15)(12,18)(13,22)(17,26)(19,28)(20,27)(23,29)(24,35)(30,32)(31,34)(33,36);(5,7,10,18,11,20,9,19)(6,12,14,27,15,28,25,16)(8,17,23,31,24,33,22,32)(13,26,21,34,29,36,35,30);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,21,15,29,25,35)(7,17,19,32,20,33,18,31)(12,26,27,34,28,36,16,30)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10944, #354114, #354318, … |
| maximal normal subgroup of | C2×#10944, #354114, #354318, … |
| maximal quotient of | #110112, #110113, #110114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3932);