Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,19)(17,20);(6,10)(9,12)(13,15,17,16)(14,18,20,19);(14,19)(15,16)(18,20)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,d^c=dh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22584, #6379686, #6379702, … |
| maximal normal subgroup of | C2×#22584, #6379686, #6379702, … |
| maximal quotient of | #224960, #224984, #225008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15572);