Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | Aut(G) = C2×#6346637 (order 1024) · Inn(G) = #1505 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1505 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | #469 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
152 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 47 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 — 3 generators on 32 points:
(1,2)(3,8)(4,16,19,7)(5,10)(6,12)(9,17,21,30)(11,18)(13,23)(14,28,29,20)(15,25)(22,26,31,32)(24,27);(2,7)(3,13)(4,17)(5,18)(8,22)(9,11)(10,21)(12,16)(14,29)(15,27)(19,30)(20,23)(24,28)(25,31);(1,3,5,13,6,15,18,27)(2,8,10,23,12,25,11,24)(4,14,17,26,19,29,30,32)(7,20,9,22,16,28,21,31)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,d^a=df,g^a=gh,c^b=cf,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11769, #481728, #481729, … |
| maximal normal subgroup of | C2×#11769, #481728, #481729, … |
| maximal quotient of | #124370, #124371, #124463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4757);