Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #386 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,17,19,9)(5,10)(6,20)(7,11)(8,15)(13,26,29,32)(14,23)(16,27)(18,25,21,24)(22,28,31,30);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,20,27)(4,13,21,30,19,29,18,28)(9,22,24,26,17,31,25,32);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,26)(14,28)(15,30)(16,29)(20,25)(23,31)(27,32);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,12,23,27)(9,24,17,25)(13,28,29,30)(22,26,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,f^b=fh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #146653, #146658, #148468, … |
| maximal normal subgroup of | #146653, #146658, #148468, … |
| maximal quotient of | #196234, #196237, #196238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13997);