Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,17)(16,19)(18,20);(1,3,4,2)(14,15)(18,19);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,19,20,18);(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=e,d^2=ef,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| maximal subgroup of | #9380, #9382, C2×#2336, … |
| maximal normal subgroup of | #9380, #9382, C2×#2336, … |
| maximal quotient of | #13781, #13787, #13790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1068);