Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2424 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,19)(3,12,14,8,15,23,20,24)(4,16,17,26,18,25,21,9)(13,28,30,32,31,22,29,27);(1,3,6,14,7,15,5,20)(2,8,10,23,11,24,19,12)(4,13,17,30,18,31,21,29)(9,22,16,27,26,28,25,32);(1,4)(2,9)(3,13)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(14,29)(15,31)(16,19)(20,30)(23,32)(24,28);(1,2,7,11)(3,8,15,24)(4,9,18,26)(5,19,6,10)(12,14,23,20)(13,22,31,28)(16,21,25,17)(27,29,32,30)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25197, #6438902, #6438904, … |
| maximal normal subgroup of | C2×#25197, #6438902, #6438904, … |
| maximal quotient of | #283580, #283581, #283592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18185);