Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23990 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2404 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
564 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(1,2,6,10)(3,7,13,21)(4,8,14,22)(5,16,18,9)(11,24,17,19)(12,26,27,20)(15,29,28,23)(25,31,30,32);(3,11,13,17)(7,19,21,24)(12,25,27,30)(20,31,26,32);(1,3)(2,7)(4,12)(5,17)(6,13)(8,20)(9,24)(10,21)(11,18)(14,27)(15,30)(16,19)(22,26)(23,32)(25,28)(29,31);(1,4)(2,8)(3,12)(5,15)(6,14)(7,20)(9,23)(10,22)(11,25)(13,27)(16,29)(17,30)(18,28)(19,31)(21,26)(24,32)Polycyclic presentation:
2^8|a^2=h,b^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,e^c=eh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #508107, #508109, #508111, … |
| maximal normal subgroup of | #508107, #508109, #508111, … |
| maximal quotient of | #251674, #251675, #251676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16980);