Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,15,18,21,19,22)(16,23,26,20,25,28,24,27);(5,7,8,12)(6,9,10,11)(13,15,17,21,18,22,19,14)(16,20,26,28,25,27,24,23);(2,4)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28);(6,10)(9,11)(13,17,18,19)(14,15,21,22)(16,24,25,26)(20,23,27,28)Polycyclic presentation:
2^8|a^2=fh,b^2=egh,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28352, #6459714, #6459734, … |
| maximal normal subgroup of | C2×#28352, #6459714, #6459734, … |
| maximal quotient of | #319104, #319115, #319126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21340);