Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | Aut(G) = #72338516 (order 4096) · Inn(G) = #2380 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2380 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
353 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 353 conjugacy classes of subgroups — too many to draw. Only its 133 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 — 4 generators on 32 points:
(1,2)(3,7,13,23)(4,15,18,25)(5,11)(6,10)(8,19,26,28)(9,17)(12,24)(14,22,20,27)(16,21)(29,32)(30,31);(1,3)(2,7)(4,16)(5,13)(6,20)(8,24)(9,22)(10,27)(11,23)(12,28)(14,17)(15,21)(18,30)(19,29)(25,31)(26,32);(1,4,17,28,5,18,6,19)(2,8,10,15,11,26,9,25)(3,12,20,16,13,29,14,30)(7,21,22,24,23,31,27,32);(2,9,11,10)(3,13)(7,22,23,27)(8,25,26,15)(12,29)(14,20)(16,30)(21,24,31,32)Polycyclic presentation:
2^8|a^2=g,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cef,d^b=de,f^b=fh,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152483, #152610, #152611, … |
| maximal normal subgroup of | #152483, #152610, #152611, … |
| maximal quotient of | #232125, #232126, #232127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16350);