Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 131072 |
| 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 | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1173 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,9)(7,12,11,8)(13,14)(15,18)(16,19)(17,22)(20,25)(21,23)(24,27)(26,28);(5,7)(6,8)(9,12)(10,11)(14,15)(16,23)(20,22)(24,28);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(5,9,10,6)(7,12,11,8)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=fh,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #208565, #208567, #208629, … |
| maximal normal subgroup of | #208565, #208567, #208629, … |
| maximal quotient of | #516418, #516604, #516717, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28298);