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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
527 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)(13,14,19,22,18,23,17,27)(15,24,25,21,16,20,26,28);(5,7,8,12)(6,9,10,11)(13,15,17,26,18,16,19,25)(14,20,27,21,23,24,22,28);(2,4)(13,16,18,15)(14,21,23,28)(17,26,19,25)(20,22,24,27);(6,10)(9,11)(13,17,18,19)(14,22,23,27)(15,25,16,26)(20,21,24,28)Polycyclic presentation:
2^8|a^2=f,b^2=eg,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×#28356, #6459731, #6459741, … |
| maximal normal subgroup of | C2×#28356, #6459731, #6459741, … |
| maximal quotient of | #319353, #319369, #319384, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21344);