Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | Aut(G) = #72261435 (order 2048) · Inn(G) = #555 · Out(G) = #120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2572 |
| prime radical | #575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,6,11,7,12,5,10)(3,13,17,24,15,28,16,8)(4,18,20,27,21,9,22,26)(14,25,19,32,30,29,31,23);(1,3)(2,8,12,24)(4,19,21,31)(5,16)(6,17)(7,15)(9,25)(10,28,11,13)(14,20,30,22)(18,29)(23,27)(26,32);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,26)(11,27)(12,18)(13,29)(15,30)(16,19)(17,31)(24,32)(25,28);(2,10,12,11)(3,15)(8,13,24,28)(9,26,18,27)(14,30)(16,17)(19,31)(23,29,32,25)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=f,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151362, #151363, C2×#29233, … |
| maximal normal subgroup of | #151362, #151363, C2×#29233, … |
| maximal quotient of | #362388, #365996, #366270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22221);