Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,3)(4,7,8,6)(11,15)(14,16)(17,18)(19,23)(20,21)(22,24);(2,3)(6,7)(10,13)(11,15)(18,20)(22,23);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,24,23);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97187, #97332, #172353, … |
| maximal normal subgroup of | #97187, #97332, #172353, … |
| maximal quotient of | #557107, #557112, #557402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38993);