Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 | #2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
502 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)(7,9)(8,10)(11,12)(13,14,18,17)(15,19,22,25)(16,20,24,26)(21,27,28,23);(2,4)(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,23)(17,25)(18,22)(20,21)(24,27)(26,28);(6,10)(9,12)(13,16)(14,20)(15,21)(17,26)(18,24)(19,27)(22,28)(23,25);(2,4)(13,14,18,17)(15,19,22,25)(16,20,24,26)(21,27,28,23)Polycyclic presentation:
2^8|a^2=h,b^2=f,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24870, #6438284, #6438305, … |
| maximal normal subgroup of | C2×#24870, #6438284, #6438305, … |
| maximal quotient of | #279949, #279950, #279953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17858);