Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2403 · Out(G) = C22×D4 |
| 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 | |
842 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 842 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)(3,13)(4,17)(5,11)(6,9)(7,12)(8,15)(10,18)(14,22)(16,24)(19,26)(20,25)(21,23)(27,28)(29,32)(30,31);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,22)(11,13)(12,23)(17,27)(18,30)(19,28)(20,29)(25,31)(26,32);(2,9)(5,6)(8,13)(10,25)(11,12)(15,21)(17,26)(18,19)(22,27)(23,24)(28,30)(31,32);(1,4,7,20)(2,10,12,26)(3,14,16,29)(5,19,6,18)(8,22,23,32)(9,25,11,17)(13,27,24,31)(15,30,21,28)Polycyclic presentation:
2^8|d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gh,c^b=cg,e^b=eh,e^c=eh,f^c=fh,e^d=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501189, #501191, #501197, … |
| maximal normal subgroup of | #501189, #501191, #501197, … |
| maximal quotient of | #244595, #244596, #244597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16872);