Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
729 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 729 conjugacy classes of subgroups — too many to draw. Only its 141 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)(3,11)(4,16)(5,20)(6,10)(7,14)(8,18)(9,17)(12,25)(13,26)(15,24)(19,29)(21,28)(22,30)(23,27)(31,32);(1,3,5,14,6,15,17,25)(2,7,9,11,10,12,20,24)(4,13,18,27,19,28,30,32)(8,21,16,23,22,26,29,31);(1,2)(3,12,15,7)(4,8)(5,9)(6,10)(11,14,24,25)(13,26,28,21)(16,18)(17,20)(19,22)(23,27,31,32)(29,30);(1,4,17,30,6,19,5,18)(2,8,20,29,10,22,9,16)(3,13,25,32,15,28,14,27)(7,21,24,31,12,26,11,23)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142616, #142618, #142620, … |
| maximal normal subgroup of | #142616, #142618, #142620, … |
| maximal quotient of | #299196, #299198, #299201, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18787);