Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) = #72267100 (order 12288) · Inn(G) = C22×C22⋊C4 · Out(G) = D4×S4 |
| 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 | C2×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16);(2,6)(4,11)(7,12)(10,15);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16);(3,9)(7,12)(10,15)(14,16)Polycyclic presentation:
2^7|a^2=f,f^2=g,b^a=be,e^a=eg,f^b=fg,d^c=dg41 conjugacy classes — view the full character table.
| Fitting subgroup of | #2888⋊C3, #11711619, #11711621, … |
| maximal subgroup of | #13097, #13099, #21778, … |
| maximal normal subgroup of | #13097, #13099, #21778, … |
| maximal quotient of | #17350, #17353, #17367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1620);