Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) = C23×#1202 (order 8192) · Inn(G) = C24 · Out(G) = C23×D42 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3475 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,17,14,11,15,19)(12,16,21,23,20,18,22,24);(2,5)(6,8);(9,11,13,19,14,10,15,17)(12,18,21,24,20,16,22,23);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(12,20)(16,18)(21,22)(23,24)Polycyclic presentation:
2^8|a^2=g,c^2=g,d^2=f,g^2=h,b^a=bf,e^d=eh100 conjugacy classes — view the full character table.
| maximal subgroup of | #129541, #210266, #210267, … |
| maximal normal subgroup of | #129541, #210266, #210267, … |
| maximal quotient of | #6369997, #6370003, #6370038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53203);