Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | Aut(G) = #72217073 (order 3072) · Inn(G) = C2×#120 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#120 |
| commutator, Frattini, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15);(2,8)(6,13)(7,14)(12,15)Polycyclic presentation:
2^7|b^a=be,c^a=cf,d^a=dg,f^b=fg,e^c=eg29 conjugacy classes — view the full character table.
| Fitting subgroup of | #3026⋊C3, #11714170, #11714173, … |
| maximal subgroup of | #23895, #23896, #23996, … |
| maximal normal subgroup of | #23895, #23896, #23996, … |
| maximal quotient of | #18284, #18285, #18286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1758);