Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | #3115 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,18)(19,24)(20,25)(21,23)(22,26)(27,32)(28,30)(29,31);(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,32)(26,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31);(2,8)(6,9)(7,16)(14,15)(17,20,21,22)(18,26,23,25)(19,27,28,29)(24,31,30,32)Polycyclic presentation:
2^8|b^2=fh,d^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #249815, #249816, #249837, … |
| maximal normal subgroup of | #249815, #249816, #249837, … |
| maximal quotient of | #6386268, #6388039, #6388040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54866);