Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C25 · Out(G) = C24×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 | |
507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,14,15)(11,12,16,19)(13,21,22,17)(18,24,23,20);(10,15)(12,19)(17,21)(20,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24);(1,3,4,7)(2,6,5,8)(9,13)(10,17)(11,18)(12,20)(14,22)(15,21)(16,23)(19,24)Polycyclic presentation:
2^7|a^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8415, #17653, #17680, … |
| maximal normal subgroup of | #8415, #17653, #17680, … |
| maximal quotient of | #34191, #34229, #34231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2247);