Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,17,22)(15,20,25,21,16,26,27,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,25,19,16,17,27)(14,20,23,21,24,26,22,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,19)(17,27)(18,25)(20,23)(22,26)(24,28);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,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×#25241, #6441735, #6441749, … |
| maximal normal subgroup of | C2×#25241, #6441735, #6441749, … |
| maximal quotient of | #285884, #285886, #285892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18229);