Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
99 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,21)(19,22,24,23);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,20,24)(18,22,21,23);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,21)(19,23)(22,24)Polycyclic presentation:
2^7|a^2=dg,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cf,f^b=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2101, #23744, #23746, … |
| maximal normal subgroup of | C2×#2101, #23744, #23746, … |
| maximal quotient of | #10689, #11208, #11268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 833);