Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
413 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14)(15,23)(16,27)(17,21)(18,32)(19,25)(20,30)(22,31)(24,38)(26,34)(28,42)(29,41)(33,43)(35,39)(36,44)(37,40);(1,3,4,2)(5,7,10,6)(8,11,9,12)(13,15)(14,19)(16,28)(17,25)(18,26)(20,35)(21,23)(22,34)(24,29)(27,38)(30,43)(31,40)(32,37)(33,36)(39,44)(41,42);(5,8,10,9)(6,11,7,12)(13,16,18,30,17,29,31,44)(14,20,22,27,21,36,32,41)(15,24,26,39,25,28,40,43)(19,33,34,42,23,35,37,38);(5,9,10,8)(6,11,7,12)(14,21)(15,25)(20,36)(22,32)(24,28)(26,40)(27,41)(39,43)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=efh,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #139113, #139115, #149714, … |
| maximal normal subgroup of | #139113, #139115, #149714, … |
| maximal quotient of | #219375, #219381, #219406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15265);