Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | Aut(G) = #72286320 (order 2048) · Inn(G) = #354 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #354 |
| commutator | C42 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| socle | C2 | #1393 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 32 points:
(1,2,4,9)(3,12,14,29)(5,10,19,27)(6,11,17,28)(7,21,22,31)(8,18,25,20)(13,24,30,26)(15,23,16,32);(1,3)(2,7)(4,16,17,21)(5,15)(6,13)(8,24)(9,26,28,29)(10,12)(11,23)(14,19,30,18)(20,31)(22,25,32,27)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=gh,f^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,c^b=ch,d^b=dfg,d^c=dgh,e^d=eh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7503, #151864, #151873, … |
| maximal normal subgroup of | C2×#7503, #151864, #151873, … |
| maximal quotient of | #94104, #94156, #94157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 491);