Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
673 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)(13,14,18,22)(15,19,25,28)(16,20,17,21)(23,26,24,27);(6,9)(10,12)(13,15,18,25)(14,19,22,28)(16,23,17,24)(20,26,21,27);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(16,17)(19,26)(21,22)(23,24)(27,28);(14,21)(16,17)(19,27)(20,22)(23,24)(26,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,f^c=fh,g^c=gh,f^d=fh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6470090, #6470091, #6470105, … |
| maximal normal subgroup of | #6470090, #6470091, #6470105, … |
| maximal quotient of | #349957, #349958, #349959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25999);