Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1265 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)(7,11)(10,12)(13,14,18,22)(15,21,17,19)(16,24,25,20)(23,27,26,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,22,21)(16,23,25,26)(20,27,24,28);(1,2)(3,4)(6,9)(10,12)(13,16)(14,20)(15,23)(17,26)(18,25)(19,27)(21,28)(22,24);(5,6)(7,10)(8,9)(11,12)(13,17,18,15)(14,21,22,19)(16,23,25,26)(20,27,24,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,25)(26,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #111986, #112315, #115355, … |
| maximal normal subgroup of | #111986, #112315, #115355, … |
| maximal quotient of | #556336, #556343, #556355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30429);