Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,22,27)(19,29,32,24)(20,33,35,42)(21,28,23,26)(25,37,40,34)(30,44,46,48)(31,39,36,41)(38,47,43,45);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,22,32)(18,24,27,29)(20,30,35,46)(21,36,23,31)(25,38,40,43)(26,41,28,39)(33,48,42,44)(34,47,37,45);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16);(2,10)(7,11)(8,9)(15,16)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=fh,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152150, #249736, #249754, … |
| maximal normal subgroup of | #152150, #249736, #249754, … |
| maximal quotient of | #6386336, #6388248, #6388310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54924);