Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,12,17)(7,18,22,34)(8,9,14,27)(10,16,11,15)(13,29,33,21)(19,20,23,36)(24,30,28,26)(25,35,31,32);(5,7,12,22)(6,13,17,33)(8,23,14,19)(9,25,27,31)(10,21,11,29)(15,18,16,34)(20,30,36,26)(24,35,28,32);(2,4)(5,8,12,14)(6,9,17,27)(7,19,22,23)(10,28,11,24)(13,31,33,25)(15,30,16,26)(18,20,34,36)(21,35,29,32);(5,9)(6,14)(7,20)(8,17)(10,26)(11,30)(12,27)(13,32)(15,28)(16,24)(18,23)(19,34)(21,31)(22,36)(25,29)(33,35);(5,10,12,11)(6,15,17,16)(7,21,22,29)(8,24,14,28)(9,26,27,30)(13,18,33,34)(19,32,23,35)(20,31,36,25)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,c^b=ch,d^b=df,f^b=fh,e^c=eh,f^d=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #130788, #130793, #131056, … |
| maximal normal subgroup of | #130788, #130793, #131056, … |
| maximal quotient of | #6359119, #6359128, #6359138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54079);