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 | #2535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1097 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)(5,6,10,9)(7,12,11,8)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(5,7)(6,8)(9,12)(10,11)(14,18)(16,24)(19,21)(22,28);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(5,6,10,9)(7,8,11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(5,9,10,6)(7,12,11,8)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=fh,c^2=g,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 | #6445226, #6445228, #6445241, … |
| maximal normal subgroup of | #6445226, #6445228, #6445241, … |
| maximal quotient of | #515347, #515359, #515477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28624);