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 | |
1162 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)(6,9)(7,11)(13,14)(15,17)(16,24)(18,19)(20,23)(21,27)(22,28)(25,26);(5,6,8,9)(7,10,11,12)(14,19)(17,20)(21,24)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,19)(18,23)(21,26)(22,27)(24,28);(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,21,19,24)(15,22,23,25)(17,26,20,28);(13,17,18,20)(14,15,19,23)(16,26,27,28)(21,22,24,25)Polycyclic presentation:
2^8|b^2=h,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 | #133114, #214913, #247789, … |
| maximal normal subgroup of | #133114, #214913, #247789, … |
| maximal quotient of | #544437, #545020, #545384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35776);