Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(10,12)(11,15)(18,21)(19,23);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,16,15,14);(2,5)(6,8)(11,15)(14,16)(17,18,20,21)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,18)(19,22)(20,21)(23,24);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=f,c^2=g,d^2=f,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 | #132533, #132565, #487321, … |
| maximal normal subgroup of | #132533, #132565, #487321, … |
| maximal quotient of | #582238, #582449, #582629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45974);