Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72286214 (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, socle | C22 | #354 |
| commutator | C42 | C42 |
| Frattini, cosocle | C2×#124 | C22 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
143 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,6)(7,17,19,31)(8,22,23,29)(9,15,10,14)(11,16)(12,18,26,34)(13,25)(20,32,21,33)(24,30)(27,35,28,36);(1,2,3,4)(5,7,9,20,11,21,10,19)(6,12,14,26,16,28,15,27)(8,18,24,34,23,36,25,35)(13,17,29,31,30,33,22,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,e^2=h,f^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fg,g^a=gh,d^b=dfgh,d^c=dgh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7502, #151860, #151861, … |
| maximal normal subgroup of | C2×#7502, #151860, #151861, … |
| maximal quotient of | #94103, #94154, #94155, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 490);