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 |
| prime radical | #2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1151 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)(8,10)(9,12)(13,14)(15,23)(16,25)(17,22)(18,19)(20,26)(21,27)(24,28);(5,6,8,10)(7,9,12,11)(14,19)(17,28)(20,23)(21,25);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,26)(14,20,19,23)(16,22,27,24)(17,25,28,21);(5,7,8,12)(6,9,10,11)(13,16,18,27)(14,21,19,25)(15,24,26,22)(17,20,28,23);(13,17,18,28)(14,22,19,24)(15,25,26,21)(16,20,27,23)Polycyclic presentation:
2^8|b^2=h,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 | #132390, #248425, #248426, … |
| maximal normal subgroup of | #132390, #248425, #248426, … |
| maximal quotient of | #544517, #545078, #545502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35796);