Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 262144 |
| 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 | |
1185 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)(6,9)(10,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27);(1,2)(3,4)(14,16)(15,22)(20,24)(23,28);(5,6,8,9)(7,10,11,12)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27);(5,7,8,11)(6,10,9,12)(13,16,18,14)(15,21,22,19)(17,24,25,20)(23,27,28,26);(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=g,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 | #111966, #112292, #112864, … |
| maximal normal subgroup of | #111966, #112292, #112864, … |
| maximal quotient of | #516545, #516549, #516697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28313);