Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) = #72267084 (order 24576) · Inn(G) = C22×C22⋊C4 · Out(G) = #73026 |
| 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 | C2×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,8)(5,9)(6,10)(12,25,28,19)(13,26,30,20)(15,21)(16,22)(17,23)(18,24)(27,32,29,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,27)(16,28)(17,29)(18,30)(21,31)(22,25)(23,32)(24,26);(1,4,6,16)(2,8,10,22)(3,12,14,28)(5,17,18,15)(7,19,11,25)(9,23,24,21)(13,29,30,27)(20,32,26,31);(1,5,6,18)(2,9,10,24)(3,13,14,30)(4,15,16,17)(7,20,11,26)(8,21,22,23)(12,27,28,29)(19,31,25,32)Polycyclic presentation:
2^7|a^2=f,c^2=g,d^2=g,b^a=be,e^a=eg,f^b=fg,d^c=dg41 conjugacy classes — view the full character table.
| Fitting subgroup of | #2884⋊C3, #11711622, #11711623, … |
| maximal subgroup of | #13101, #13103, #21783, … |
| maximal normal subgroup of | #13101, #13103, #21783, … |
| maximal quotient of | #17352, #17353, #17371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1616);