Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7481 |
| automorphism | Aut(G) = #72236721 (order 2048) · Inn(G) = #1390 · Out(G) = C24 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1390 |
| commutator | C2×Q8 | C42 |
| Frattini, prime radical, cosocle | #532 | C22 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 133 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:
(1,2,4,9,6,12,21,28)(3,13,15,7,18,23,31,25)(5,11,19,8,22,10,20,26)(14,29,17,27,32,24,16,30);(1,3,5,16,6,18,22,17)(2,7,10,24,12,25,11,29)(4,14,19,31,21,32,20,15)(8,13,9,27,26,23,28,30)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fh,g^b=gh,d^c=dgh,f^c=fh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7484, #151788, #151825, … |
| maximal normal subgroup of | C2×#7484, #151788, #151825, … |
| maximal quotient of | #94085, #94128, #94141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 472);