Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C2×#612424 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C42 | C23 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | #121 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^7|a^2=e,c^2=f,d^2=f,e^2=g,b^a=bd,d^a=df,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8489, #8490, #8514, … |
| maximal normal subgroup of | #8489, #8490, #8514, … |
| maximal quotient of | #7656, #7657, #7658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 214);