Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(5,8)(10,13)(11,15)(17,18,21,23)(19,20,24,22);(1,2)(3,5)(4,6)(7,8)(10,13)(14,16)(17,19,21,24)(18,22,23,20);(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,14,15,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,20)(21,24)(22,23);(1,3,4,7)(2,5,6,8)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7201543, #7201547, #7201607, … |
| maximal normal subgroup of | #7201543, #7201547, #7201607, … |
| maximal quotient of | #582167, #582389, #582399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46085);