Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| 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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,14,19)(11,20,23,16)(12,24,27,35)(13,18,15,29)(17,30,36,26)(21,37,25,40)(22,33,28,32)(31,39,34,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,25)(13,28)(14,23)(15,22)(17,34)(18,33)(19,20)(21,27)(24,37)(26,38)(29,32)(30,39)(31,36)(35,40);(9,12)(10,17)(11,21)(13,26)(14,27)(15,30)(16,31)(18,35)(19,36)(20,34)(22,38)(23,25)(24,29)(28,39)(32,40)(33,37);(2,5)(6,8)(9,13,14,15)(10,18,19,29)(11,22,23,28)(12,26,27,30)(16,32,20,33)(17,35,36,24)(21,38,25,39)(31,40,34,37);(1,3,4,7)(2,6,5,8)(11,23)(16,20)(21,25)(22,28)(31,34)(32,33)(37,40)(38,39)Polycyclic presentation:
2^8|a^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,g^a=gh,c^b=ch,d^b=df,e^b=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #131806, #131820, #489323, … |
| maximal normal subgroup of | #131806, #131820, #489323, … |
| maximal quotient of | #6379571, #6380399, #6380889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54489);