Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #117 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #432 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,9,17,25,32,11,16,27,13,26,14,8,23)(7,18,21,33,20,12,19,28,22,30,34,36,35,29,24,31);(1,2,3,4)(5,7)(6,12,16,29)(8,24)(9,20)(10,21)(11,22)(13,31,15,28)(14,18,17,30)(19,25)(23,33,32,36)(26,35)(27,34);(5,8,9,10,11,25,26,27)(6,13,14,32,16,15,17,23)(7,19,20,34,22,24,35,21)(12,28,18,33,29,31,30,36)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=d,d^2=h,e^2=g,g^2=h,b^a=bd,c^a=cg,d^a=dh,f^a=fh,c^b=ch,e^c=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93355, #93649, #124770, … |
| maximal normal subgroup of | #93355, #93649, #124770, … |
| maximal quotient of | #128212, #128219, #128224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5584);