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 | #535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,23,26)(19,24,29,40)(20,31,33,42)(21,27,22,35)(25,36,41,32)(28,43,44,48)(30,38,34,39)(37,45,47,46);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,24)(20,28)(21,34)(22,30)(23,29)(25,37)(26,40)(27,39)(31,43)(32,46)(33,44)(35,38)(36,45)(41,47)(42,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,45)(31,35)(34,46)(38,43)(39,48)(40,47);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(18,26)(19,29)(25,41)(27,35)(28,44)(30,34)(31,42)(45,46);(2,10)(7,11)(8,9)(15,16)(17,21,23,22)(18,27,26,35)(19,30,29,34)(20,32,33,36)(24,38,40,39)(25,42,41,31)(28,45,44,46)(37,43,47,48)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,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 | #152146, #152152, #249806, … |
| maximal normal subgroup of | #152146, #152152, #249806, … |
| maximal quotient of | #6386339, #6388240, #6388258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54931);