Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| 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) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,21)(15,16,19,24)(17,20,26,28)(22,23,25,27);(6,9)(10,11)(13,15,18,19)(14,16,21,24)(17,25,26,22)(20,27,28,23);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,22)(18,19)(20,23)(21,24)(25,26)(27,28);(5,7,8,12)(6,10,9,11)(13,16)(14,19)(15,21)(17,23)(18,24)(20,25)(22,28)(26,27);(13,17,18,26)(14,20,21,28)(15,22,19,25)(16,23,24,27)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132361, #248368, #248369, … |
| maximal normal subgroup of | #132361, #248368, #248369, … |
| maximal quotient of | #544516, #545080, #545503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35875);