Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,9,10)(7,11,12,8)(14,15)(18,19)(21,22)(23,27)(24,25)(26,28);(6,10)(7,12)(13,14,17,15)(16,18,20,19)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,15,17,14)(16,18,20,19)(21,23)(22,27)(24,26)(25,28);(5,8,9,11)(6,12,10,7)(13,16,17,20)(14,18,15,19)(21,24,25,22)(23,27,28,26)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#16702, #611436, #611546, … |
| maximal normal subgroup of | C2×#16702, #611436, #611546, … |
| maximal quotient of | #155688, #155787, #155827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9690);