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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,9,7)(8,11,12,10)(14,17)(18,19)(21,22,24,26)(23,25,27,28);(6,7)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,25,26,28);(1,2,4,3)(5,7,9,6)(8,10,12,11)(13,14)(15,19)(16,17)(18,20)(21,23)(22,25)(24,27)(26,28);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,20)(17,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=g,b^2=efh,c^2=ef,d^2=ef,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×#16790, #611530, #611550, … |
| maximal normal subgroup of | C2×#16790, #611530, #611550, … |
| maximal quotient of | #155704, #155723, #155796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9778);