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 | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1016 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,16,24,34)(12,18,26,36)(13,19,14,29)(17,28,37,25)(21,38,22,31)(23,32,27,33)(30,39,35,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,27)(14,23)(15,24)(17,35)(18,31)(19,33)(20,34)(21,26)(25,39)(28,40)(29,32)(30,37)(36,38);(9,12)(10,17)(11,21)(13,25)(14,28)(15,26)(16,30)(18,29)(19,36)(20,37)(22,24)(23,39)(27,40)(31,33)(32,38)(34,35);(3,8)(5,7)(9,12)(10,18)(11,22)(13,28)(14,25)(15,26)(16,31)(17,29)(19,37)(20,36)(21,24)(23,39)(27,40)(30,33)(32,35)(34,38);(2,6)(5,7)(9,13,15,14)(10,19,20,29)(11,23,24,27)(12,25,26,28)(16,32,34,33)(17,36,37,18)(21,39,22,40)(30,38,35,31)Polycyclic presentation:
2^8|a^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #249800, #249803, #249820, … |
| maximal normal subgroup of | #249800, #249803, #249820, … |
| maximal quotient of | #6386060, #6387680, #6387682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54662);