Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18414 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
497 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2,7,4)(3,13,17,27)(5,11,6,19)(8,12,24,20)(9,22,14,15)(10,23,18,21)(16,29,31,25)(26,32,30,28);(1,3,7,17)(2,9,4,14)(5,15,6,22)(8,16,24,31)(10,25,18,29)(11,27,19,13)(12,26,20,30)(21,28,23,32);(1,4,7,2)(3,14,17,9)(5,10,6,18)(8,20,24,12)(11,23,19,21)(13,28,27,32)(15,25,22,29)(16,30,31,26);(1,5,7,6)(2,10,4,18)(3,15,17,22)(8,21,24,23)(9,25,14,29)(11,20,19,12)(13,26,27,30)(16,28,31,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244503, #244504, #244509, … |
| maximal normal subgroup of | #244503, #244504, #244509, … |
| maximal quotient of | #169622, #169632, #169685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11413);