Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2403 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13,18,26)(4,19,22,29)(5,23,6,11)(8,17,27,16)(9,20,10,21)(14,30,31,24)(15,25,32,28);(1,3)(2,8)(4,15)(5,17)(6,16)(7,18)(9,28)(10,25)(11,26)(12,27)(13,23)(14,20)(19,24)(21,31)(22,32)(29,30);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,24)(10,12)(11,29)(13,25)(15,16)(17,32)(18,31)(19,23)(26,28)(27,30);(1,4)(2,10)(3,15)(5,21)(6,20)(7,22)(8,25)(9,12)(11,29)(13,24)(14,16)(17,31)(18,32)(19,23)(26,30)(27,28)Polycyclic presentation:
2^8|a^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,e^b=eh,g^b=gh,f^c=fh,g^c=gh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23894, #6437592, #6437593, … |
| maximal normal subgroup of | C2×#23894, #6437592, #6437593, … |
| maximal quotient of | #244524, #244525, #244527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16882);