Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,16,24)(12,21,22,20)(13,19,23,17);(2,5)(6,8)(9,11)(10,15)(12,19)(13,20)(14,16)(17,22)(18,24)(21,23);(9,12,13,15,14,22,23,24)(10,16,17,21,18,11,19,20);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,20)(14,16)(17,22)(18,24)(21,23);(11,16)(12,22)(15,24)(20,21)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,e^b=eh,e^c=eh,e^d=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #249735, #249736, #249737, … |
| maximal normal subgroup of | #249735, #249736, #249737, … |
| maximal quotient of | #6371191, #6372141, #6372456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53711);