Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6,8,12,9,13,16,11)(7,10,15,17,14,18,20,19);(1,2,4,3)(5,7,8,15,9,14,16,20)(6,10,12,17,13,18,11,19);(6,11)(7,14)(8,16)(10,17)(12,13)(18,19);(1,3,4,2)Polycyclic presentation:
2^8|a^2=f,b^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #285115, #285117, #285132, … |
| maximal normal subgroup of | #285115, #285117, #285132, … |
| maximal quotient of | #287739, #287746, #287751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18392);