Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #127 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
284 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,4)(3,9,11,16)(5,13,15,8)(7,14,12,10)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,28,32,26);(1,3,6,11)(2,7,4,12)(5,14,15,10)(8,16,13,9)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(1,4,6,2)(3,9,11,16)(5,8,15,13)(7,14,12,10)Polycyclic presentation:
2^8|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | #153632, C2×#20117, #6359453, … |
| maximal normal subgroup of | #153632, C2×#20117, #6359453, … |
| maximal quotient of | #222393, #222553, #226115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13105);