Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,15,9,25)(7,16,20,26,17,31,19,12)(8,13,23,29,24,30,22,36)(18,32,34,28,21,35,33,27);(5,7,9,19,11,17,10,20)(6,12,14,16,15,26,25,31)(8,21,22,34,24,18,23,33)(13,28,29,35,30,27,36,32);(2,4)(7,17)(12,26)(16,31)(18,21)(19,20)(27,28)(32,35)(33,34);(5,8,10,23,11,24,9,22)(6,13,14,29,15,30,25,36)(7,18,20,34,17,21,19,33)(12,27,16,32,26,28,31,35)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128806, #128807, #134460, … |
| maximal normal subgroup of | #128806, #128807, #134460, … |
| maximal quotient of | #210808, #210809, #210810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14538);