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 | |
528 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)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24);(2,5)(6,8)(10,11)(12,19)(16,18)(20,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^7|c^2=f,d^2=fg,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 | #8489, #18248, #18249, … |
| maximal normal subgroup of | #8489, #18248, #18249, … |
| maximal quotient of | #34381, #34388, #34401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2292);