Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16,14,17,12,21)(11,18,20,15,19,24,22,23);(1,3,4,8)(2,5,6,7)(11,19)(15,23)(18,24)(20,22);(3,8)(5,7)(9,11)(10,15)(12,22)(13,20)(14,19)(16,24)(17,23)(18,21);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=g,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #107810, #107813, #107973, … |
| maximal normal subgroup of | #107810, #107813, #107973, … |
| maximal quotient of | #215183, #215184, #215185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14804);