Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) = C2×#7622880 (order 1024) · Inn(G) = C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊C4 |
| commutator | C22 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | C4×C22⋊C4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,17,23)(3,11,14,25,13,7,18,20)(4,8,15,21,16,22,28,31)(12,24,27,32,26,19,29,30);(1,3)(2,7,10,11)(4,12)(5,14)(6,13)(8,19,22,24)(9,20,23,25)(15,27)(16,26)(17,18)(21,30,31,32)(28,29);(1,4,5,15,6,16,17,28)(2,8,9,21,10,22,23,31)(3,12,14,27,13,26,18,29)(7,19,20,30,11,24,25,32)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=e,e^2=g,b^a=bd,f^a=fg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7207, #7208, #12613, … |
| maximal normal subgroup of | #7207, #7208, #12613, … |
| maximal quotient of | #8156, #8157, #8158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 489);