Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C22×#6342371 (order 2048) · Inn(G) = C25 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(10,14)(11,17)(12,20)(22,24);(2,5)(6,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)Polycyclic presentation:
2^7|b^2=f,d^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df38 conjugacy classes — view the full character table.
| maximal subgroup of | #8488, #18226, #18227, … |
| maximal normal subgroup of | #8488, #18226, #18227, … |
| maximal quotient of | #34359, #34376, #34386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2267);