Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) = #72241374 (order 8192) · Inn(G) = C22×C22⋊C4 · Out(G) = #2196 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| prime radical | C2×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×D4 ◁ 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16);(4,12)(7,14)(10,16)(13,15)Polycyclic presentation:
2^7|a^2=f,b^a=be,e^a=eg,f^b=fg,d^c=dg41 conjugacy classes — view the full character table.
| maximal subgroup of | #13100, #13102, #21776, … |
| maximal normal subgroup of | #13100, #13102, #21776, … |
| maximal quotient of | #17350, #17351, #17367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1615);