Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2403 · Out(G) = C25 |
| 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 | |
837 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 837 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,15)(5,7)(6,11)(9,16)(10,18)(12,27)(13,22)(14,24)(17,26)(19,21)(20,25)(23,28)(29,31)(30,32);(2,7)(5,18)(8,22)(9,15)(10,11)(13,19)(16,20)(21,24)(23,27)(25,26)(28,30)(31,32);(1,3,6,14)(2,8,11,24)(4,12,17,29)(5,19,18,13)(7,21,10,22)(9,23,25,32)(15,31,26,27)(16,30,20,28);(1,4)(2,9)(3,12)(5,20)(6,17)(7,15)(8,23)(10,26)(11,25)(13,30)(14,29)(16,18)(19,28)(21,31)(22,27)(24,32)Polycyclic presentation:
2^8|c^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152666, #152673, #152704, … |
| maximal normal subgroup of | #152666, #152673, #152704, … |
| maximal quotient of | #244551, #244552, #244555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16862);