Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7519 |
| automorphism | Aut(G) = C2×#512862 (order 1024) · Inn(G) = #1402 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1402 |
| commutator | C4⋊Q8 | C2×C4 |
| Frattini, cosocle | #494 | C22 |
| prime radical | #555 | C4 |
| LCS | G ▷ C4⋊Q8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ G | |
| LECS | G ▷ #494 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,5,11,17,29)(3,12,14,26,13,23,32,27)(6,10,21,30,19,31,18,28)(7,16,22,20,8,24,25,15);(2,7)(3,13)(8,11)(9,26)(10,12)(14,32)(15,20)(16,24)(22,28)(23,31)(25,30)(27,29)Polycyclic presentation:
2^8|a^2=d,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ch,d^b=deg,e^b=egh,f^b=fgh,d^c=dfg,e^c=egh,f^c=fh,e^d=egh,f^d=fh22 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7521, #153453, #153454, … |
| maximal normal subgroup of | C2×#7521, #153453, #153454, … |
| maximal quotient of | #94540, #94595, #94672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 509);