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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1023 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)(5,6,9,11)(7,8,12,10)(13,14)(15,23)(16,25)(17,22)(18,19)(20,26)(21,27)(24,28);(5,7,9,12)(6,10,11,8)(14,19)(17,28)(20,23)(21,25);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,18,26)(14,20,19,23)(16,22,27,24)(17,25,28,21);(5,8)(6,7)(9,10)(11,12)(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|a^2=h,b^2=h,c^2=g,d^2=fh,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 | #131420, #131423, #484591, … |
| maximal normal subgroup of | #131420, #131423, #484591, … |
| maximal quotient of | #544513, #545073, #545497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35968);