Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(10,11)(13,14,18,20)(15,21,23,16)(17,19,27,28)(22,26,25,24);(2,4)(5,6,9,7)(8,10,12,11)(13,15,18,23)(14,16,20,21)(17,25,27,22)(19,26,28,24);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,16)(14,15)(17,26)(18,21)(19,25)(20,23)(22,28)(24,27);(5,8,9,12)(6,10,7,11)(13,17,18,27)(14,19,20,28)(15,22,23,25)(16,24,21,26)Polycyclic presentation:
2^7|a^2=ef,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| maximal subgroup of | #10080, #10084, #10087, … |
| maximal normal subgroup of | #10080, #10084, #10087, … |
| maximal quotient of | #14629, #14631, #14634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1554);