Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
812 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 812 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,3)(5,6)(7,10)(8,11)(9,12);(1,2,4,3)(5,6)(7,11)(8,10)(9,12);(6,10)(7,8)(11,12);(1,3,4,2)Polycyclic presentation:
2^8|b^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #250071, #250077, #250080, … |
| maximal normal subgroup of | #250071, #250077, #250080, … |
| maximal quotient of | #261267, #261270, #261279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17218);