Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | C2×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
548 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)(13,14,18,26,17,22,16,21)(15,19,23,20,25,28,24,27);(5,7,8,12)(6,9,10,11)(13,15,18,23,17,25,16,24)(14,19,26,20,22,28,21,27);(2,4)(13,15,17,25)(14,20,22,27)(16,23,18,24)(19,21,28,26);(6,10)(9,11)(13,16,17,18)(14,21,22,26)(15,23,25,24)(19,20,28,27)Polycyclic presentation:
2^8|a^2=fh,b^2=egh,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28368, #6459731, #6459741, … |
| maximal normal subgroup of | C2×#28368, #6459731, #6459741, … |
| maximal quotient of | #319356, #319370, #319383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21356);