Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| 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 | #3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,20,32)(12,16,24,30)(13,26,21,17)(15,27,28,23)(22,33,34,40)(25,37,35,31)(29,38,39,36);(2,5)(6,8)(11,20)(15,28)(19,32)(22,34)(23,27)(29,39)(33,40)(36,38);(9,11,21,27,14,20,13,23)(10,15,26,32,18,28,17,19)(12,22,35,38,24,34,25,36)(16,29,37,40,30,39,31,33);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,22)(13,25)(14,24)(15,29)(17,31)(18,30)(19,33)(20,34)(21,35)(23,36)(26,37)(27,38)(28,39)(32,40);(12,24)(16,30)(22,34)(25,35)(29,39)(31,37)(33,40)(36,38)Polycyclic presentation:
2^8|a^2=h,c^2=gh,d^2=f,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129942, #129944, #481730, … |
| maximal normal subgroup of | #129942, #129944, #481730, … |
| maximal quotient of | #6371216, #6372137, #6372139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53699);