Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(2,6)(5,8)(11,18)(14,17)(19,24)(21,23);(10,15)(14,17)(16,22)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^7|b^2=g,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=cf44 conjugacy classes — view the full character table.
| maximal subgroup of | #8411, #12720, #18126, … |
| maximal normal subgroup of | #8411, #12720, #18126, … |
| maximal quotient of | #34048, #34056, #34065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2187);