Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 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)(13,14,17,21,18,22,15,19)(16,24,23,20,25,27,26,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,17)(14,19,22,21)(16,23,25,26)(20,27,28,24);(2,4)(5,8,9,12)(6,10,7,11)(14,19)(15,17)(20,27)(21,22)(23,26)(24,28);(6,7)(10,11)(13,16,18,25)(14,20,22,28)(15,23,17,26)(19,27,21,24)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6635214, #6635224, #6635242, … |
| maximal normal subgroup of | #6635214, #6635224, #6635242, … |
| maximal quotient of | #344225, #344227, #344233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24157);