Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 1024 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #546 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,19)(13,22)(14,17)(16,18)(20,24)(21,23);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,19,24,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,24)(19,22);(3,8)(5,7)(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,23,24,19)Polycyclic presentation:
2^7|a^2=e,b^2=g,d^2=g,f^2=g,b^a=be,c^a=cf,f^a=fg,c^b=cg,d^b=de,f^c=fg29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3311, #61141, #61156, … |
| maximal normal subgroup of | C2×#3311, #61141, #61156, … |
| maximal quotient of | #22424, #22494, #22568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2043);