Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) = #72232993 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator, prime radical | C23 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
102 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=f,e^2=g,b^a=bd,d^a=df,c^b=cg,e^b=ef32 conjugacy classes — view the full character table.
| maximal subgroup of | #8476, #8478, #8485, … |
| maximal normal subgroup of | #8476, #8478, #8485, … |
| maximal quotient of | #7657, #7660, #7663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 202);