Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#12067 (order 512) · Inn(G) = D16 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D16 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| socle | C2 | #357 |
| prime radical | #508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,13,26,12)(3,14,16,43,19,44,18,50)(4,20,23,51,25,53,58,52)(6,27,22,11,29,39,57,41)(8,30,31,46,33,45,42,15)(9,34,36,59,38,61,37,24)(17,48,47,62,49,63,28,35)(21,40,54,60,56,64,55,32);(1,3)(2,8)(4,21)(5,16)(6,28)(7,19)(9,35)(10,31)(11,40)(12,42)(13,33)(14,20)(15,24)(17,22)(18,26)(23,54)(25,56)(27,32)(29,47)(30,34)(36,48)(37,63)(38,62)(39,60)(41,64)(43,51)(44,53)(45,61)(46,59)(49,57)(50,52)(55,58)Polycyclic presentation:
2^7|a^2=d,c^2=ef,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=ce,e^b=ef,f^b=fg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1417, #13620, #13655, … |
| maximal normal subgroup of | C2×#1417, #13620, #13655, … |
| maximal quotient of | #7383, #7384, #7420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 149);