Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | Aut(G) = #72280587 (order 2048) · Inn(G) = C2×C42 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C22 | C2×#97 |
| prime radical | C23 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14,15,20)(11,17)(16,23,22,21)(18,24);(9,10,11,16,12,14,18,21,13,15,17,22,19,20,24,23);(2,5)(6,8)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,b^a=bg,c^a=cf,e^a=eh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #128513, #128514, #128515, … |
| maximal normal subgroup of | #128513, #128514, #128515, … |
| maximal quotient of | #126534, #126562, #128211, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4401);