Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C2×D4 · Out(G) = C210 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
594 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)(15,19)(16,23)(17,25)(18,21)(20,22)(24,27)(26,28);(1,2)(3,4)(6,9)(10,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(2,4)(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #208697, #208703, #208709, … |
| maximal normal subgroup of | #208697, #208703, #208709, … |
| maximal quotient of | #241412, #241416, #241424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16560);