Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
655 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(15,23)(16,17)(19,27)(20,21)(24,28)(25,26);(6,9)(10,12)(13,14,18,22)(15,24,25,27)(16,21,17,20)(19,23,28,26);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,26)(17,23)(18,25)(20,24)(21,27)(22,28);(6,9)(10,12)(15,25)(19,28)(23,26)(24,27)Polycyclic presentation:
2^8|b^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24047, #6440404, #6440414, … |
| maximal normal subgroup of | C2×#24047, #6440404, #6440414, … |
| maximal quotient of | #255005, #255009, #255021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17035);