Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
824 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)(3,7)(4,6)(5,8)(10,15)(11,16)(18,22)(23,24);(1,2,4,6)(3,5,8,7)(9,10)(11,12)(13,16)(14,15)(17,18,21,22)(19,23,20,24);(1,2)(3,5)(4,6)(7,8)(9,11,14,16)(10,12,15,13)(17,19,21,20)(18,23,22,24);(1,2)(3,5)(4,6)(7,8)(9,12,14,13)(10,11,15,16);(1,3,4,8)(2,5,6,7)(9,13,14,12)(10,16,15,11)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|b^2=fg,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #161609, #161893, #162389, … |
| maximal normal subgroup of | #161609, #161893, #162389, … |
| maximal quotient of | #516513, #516539, #516692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28104);