Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
976 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)(13,14,18,23)(15,20,25,19)(16,21,26,27)(17,24,28,22);(5,6)(7,9)(8,11)(10,12)(13,15,18,25)(14,19,23,20)(16,24,26,22)(17,27,28,21);(1,2,3,4)(6,7)(10,11)(13,15)(14,20)(16,22)(17,21)(18,25)(19,23)(24,26)(27,28);(5,7,9,6)(8,11,12,10)(13,16)(14,21)(15,24)(17,19)(18,26)(20,28)(22,25)(23,27);(5,8)(6,10)(7,11)(9,12)(13,17,18,28)(14,22,23,24)(15,21,25,27)(16,19,26,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132259, #245961, #245962, … |
| maximal normal subgroup of | #132259, #245961, #245962, … |
| maximal quotient of | #560465, #560486, #561561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43680);