Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(14,18)(15,21)(16,24)(26,28);(5,6,8,10)(7,9,12,11)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,7,8,12)(6,9,10,11)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28);(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^8|a^2=h,b^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #6399553, #6399557, #6399569, … |
| maximal normal subgroup of | #6399553, #6399557, #6399569, … |
| maximal quotient of | #513826, #513845, #513854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27463);