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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
483 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)(7,9)(8,10)(11,12)(13,14,18,22)(15,20,17,19)(16,24,25,27)(21,26,28,23);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,19)(15,17)(20,22)(21,24)(23,26)(27,28);(2,4)(6,10)(9,12)(13,15,18,17)(14,20,22,19)(16,23,25,26)(21,27,28,24);(13,16,18,25)(14,21,22,28)(15,23,17,26)(19,24,20,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24904, #6438182, #6438203, … |
| maximal normal subgroup of | C2×#24904, #6438182, #6438203, … |
| maximal quotient of | #279888, #279890, #279896, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17892);