Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = #72225052 (order 2048) · Inn(G) = D16 · Out(G) = C8×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D16 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4)(6,11)(7,16)(8,17)(9,20)(12,19)(13,28)(14,31)(15,25)(18,27)(21,34)(22,23)(24,29)(26,35)(30,36)(32,33);(5,6,12,26,8,13,18,24,20,29,33,28,34,35,7,11,10,15,16,23,21,30,32,31,9,14,27,36,17,22,19,25)Polycyclic presentation:
2^7|a^2=d,b^2=ceg,c^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | #7540, #7548, C2×#1424, … |
| maximal normal subgroup of | #7540, #7548, C2×#1424, … |
| maximal quotient of | #7461, #7471, #7548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 156);