Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,12)(7,9,11,8);(5,7,10,11)(6,8,12,9)(14,18)(16,24)(19,21)(22,28);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(5,8,10,9)(6,7,12,11)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(5,9,10,8)(6,11,12,7)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=fh,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #6439832, #6439836, #6439841, … |
| maximal normal subgroup of | #6439832, #6439836, #6439841, … |
| maximal quotient of | #515310, #515325, #515476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28653);