Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,9,12)(7,8,11,10)(15,19)(17,20);(1,2,3,4)(5,7,9,11)(6,10,12,8)(13,14,16,18)(15,17,19,20);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,19)(18,20);(5,8,9,10)(6,11,12,7)Polycyclic presentation:
2^7|a^2=g,b^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #10090, #10092, #10098, … |
| maximal normal subgroup of | #10090, #10092, #10098, … |
| maximal quotient of | #14260, #14261, #14273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1372);